Study Augmenting TAK-659 Action in Relapsed/Refractory AML by Addition Ofthe Proteasome Inhibitor Ixazomib

Inclusion Criteria: * Written informed consent and HIPAA authorization for release of personal health information. NOTE: HIPAA authorization may be included in the informed consent or obtained separately. * Male or female patients 18 years or older at the time of consent. * Patients must have a diagnosis of primary or secondary AML with relapsed or refractory disease (WHO update 2016, \[34\]) other than acute promyelocytic leukemia, or complex karyotype or monosomy 7 (or containing monosomy 7) for whom no standard therapies are anticipated to result in a durable remission according to the clinical judgment of the treating physician, or in a patient who refuses standard therapies. * Must have submitted for Next Generation Sequencing (NGS) testing by: 1) (preferred) having submitted a tumor sample for commercial myeloid NGS from Foundation One, Mayo, Tempus, Quest, ARUP or an institutional CLIA-certified laboratory and/or 2) obtaining a bone marrow aspirate during screening for submission to Foundation One, Mayo, Tempus, Quest, ARUP or an institutional CLIA-certified laboratory and ordered as standard of care. If bone marrow aspiration has failed, a peripheral blood sample with circulating blasts may be substituted. Screening reports on tumor cytogenetics and/or mutation assays (eg, FLT-3, and NGS) performed as part of the standard of care will be recorded in the study database or obtained at the time of screening if not previously available. * Patient's disease must be characterized for the presence/absence of Flt3ITD/TKD mutation from an FDA-approved vendor (ex. LabPMM). * In addition, patients for the phase 2 portion of the study must meet the following: * Patients, if relapsed/ refractory, must have exposure to no more than 2 prior chemotherapy regimens. Re-induction with the same regimen or stem cell transplant will not be considered a separate regimen. * Patients must not have prior exposure to any investigational Flt3 inhibitors (midostaurin or gilteritinib are allowed) * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 * Life expectancy of greater than 3 months as determined by the treating physician. * Female patients who: * Are postmenopausal for at least 1 year before the screening visit, OR * Are surgically sterile, OR * If they are of childbearing potential, agree to practice 2 effective methods of contraception at the same time, from the time of signing the informed consent through 180 days after the last dose of study drug, or * Agree to practice true abstinence, when is in line with the preferred and usual lifestyle of the subject. Periodic abstinence \[eg, calendar, ovulation, symptothermal, postovulation methods\], withdrawal, spermicides only, and lactational amenorrhea are not acceptable methods of contraception. Female and male condoms should not be used together.) * Agree not to donate eggs (ova) during the course of this study or 180 days after receiving their last dose of study drug. * Male patients, even if surgically sterilized (i.e., status post-vasectomy), who: * Agree to practice effective barrier contraception during the entire study treatment period and through 180 days after the last dose of study drug, or * Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence \[eg, calendar, ovulation, symptothermal, postovulation methods for the female partner\] and withdrawal are not acceptable methods of contraception. Female and male condoms should not be used together.) * Agree not to donate sperm during the course of this study or within 180 days after receiving their last dose of study drug. * Voluntary written consent must be given before performance of any study related procedure not part of standard medical care, with the understanding that consent may be withdrawn by the patient at any time without prejudice to future medical care * In the absence of rapid progressive disease, the interval from prior systemic anticancer treatment to time of TAK-659/Ixazomib administration should be at least 2 weeks for cytotoxic agents (other than hydroxyurea), or at least 5 half-lives for noncytotoxic agents, and patients have to have recovered from acute toxicities of these therapies. Patients who are on hydroxyurea may be included in the study and may continue on hydroxyurea for the first 21 days while participating in this study. NOTE: For patients with a white blood cell count \>50,000/uL, hydroxyurea may be used to control the level of circulating leukemic blast cell counts prior to study entry, and, if needed, concomitantly while on TAK-659 treatment during the first 21 days of the study. * Suitable venous access for the study-required blood sampling, and transfusion support. * Demonstrate adequate organ function as defined in the protocol; all screening labs to be obtained within 28 days prior to registration. Exclusion Criteria: * Clinically active central nervous system leukemia. * Female patients who are lactating and breastfeeding (NOTE: breast milk cannot be stored for future use while the mother is being treated on study) * Female patients who have a positive serum pregnancy test during the screening period or a positive urine pregnancy test on Day 1 before first dose of study drug. * Any serious medical or psychiatric illness, including drug or alcohol abuse, that could, in the investigator's opinion, potentially jeopardize the safety of the patient or interfere with the objectives of the study. * Prior treatment with investigational agents ≤ 21 days or ≤ 5 half-lives (whichever is shorter) before the first dose of study treatment. A minimum of 10 days should elapse from prior investigational therapy to initiating protocol therapy. * Persistent clinically significant toxicity from prior chemotherapy that is Grade 2 or higher by the NCI CTCAE (v5). * Receipt of HSCT within 60 days of the first dose of TAK-659/Ixazomib; clinically significant graft-versus-host disease (GVHD) requiring ongoing immunosuppressive therapy post HSCT at the time of screening (use of topical steroids for ongoing skin GVHD is permitted). * Active, systemic infection requiring intravenous (IV) antibiotic, antifungal, or antiviral therapy or other serious infection within 10 days before the first dose of study drug. * Major surgery within 14 days before the first dose of study drug and have not recovered fully from any complications from surgery. * Radiotherapy less than 2 weeks before the first dose of study treatment or have not recovered from acute toxic effects from radiotherapy. * Known human immunodeficiency virus (HIV) positive. * Known hepatitis B surface antigen positive, or known or suspected active hepatitis C infection (testing not required). * Any of the following cardiovascular conditions: * Acute myocardial infarction within 6 months before starting study drug. * Current or history of New York Heart Association Class III or IV heart failure (see Study Procedures Manual \[SPM\]). * Evidence of current, uncontrolled cardiovascular conditions including cardiac arrhythmias, angina, pulmonary hypertension, or electrocardiographic evidence of acute ischemia or active conduction system abnormalities. * Fridericia corrected QT interval (QTcF) \>450 milliseconds (msec) (men) or \>475 msec (women) on a 12-lead ECG during the Screening period. * Abnormalities on 12-lead ECG including, but not limited to, changes in rhythm and intervals that, in the opinion of the treating physician, are considered to be clinically significant. * Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance to study drugs, including difficulty swallowing tablets, diarrhea \> Grade I despite supportive therapy. * Use or consumption of any of the following medications, supplements, or foods/beverages that are inhibitors or inducers of P-gp or strong reversible inhibitors or inducers of CYP3A within the indicated timeframes below. Note that use or consumption of these substances is not permitted during the study. * Inhibitors of P-gp and/or strong reversible inhibitors of CYP3A within 5 times the inhibitor half-life (if a reasonable half-life estimate is known), or within 7 days (if a reasonable half-life estimate is unknown), before the first dose of study drug. In general, the use of these agents is not permitted during the study except in cases where an AE must be managed. See SPM for a nonexhaustive list of strong CYP3A reversible inhibitors and/ or Pgp inhibitors based on the FDA Draft DDI Guidance. * Strong CYP3A mechanism-based inhibitors or strong CYP3A inducers and/or P-gp inducers within 7 days, or within 5 times the inhibitor or inducer half-life (whichever is longer), before the first dose of study drug. However, if a patient has a strong indication for fungal prophylaxis, then the moderate CYP3A inhibitors fluconazole or isovuconazole (mold-active) is recommended for prophylaxis, and the TAK-659 dose should be reduced to 60mg, along with increased scrutiny for toxicity. Similarly, if the patient develops a fungal infection during a productive intervention with TAK-659/Ixazomib combination, isovuconazole (mold-active) should be used, along with increased scrutiny for toxicity. In general, the use of these agents is not recommended during the study except in cases where such an AE must be managed, and the intermittent schedule of TAK-659/Ixazomib combination will not prevent profound neutropenia. See SPM for a non-exhaustive list of strong CYP3A mechanism-based inhibitors or strong CYP3A inducers and/or P-gp inducers based on the FDA Draft DDI Guidance. * Grapefruit-containing food or beverages within 5 days before the first dose of study drug. * Lack of suitable venous access for the study-required blood sampling. * Diagnosed or treated for another malignancy within 2 years before study enrollment or previously diagnosed with another malignancy and have any evidence of residual disease. Patients with nonmelanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection. * Patient has Grade 3 peripheral neuropathy, or Grade 2 with pain on clinical examination during the screening period. * Known allergy to any of the study medications, their analogues, or excipients in the various formulations of any agent.