Cytokine-induced Memory-like NK Cells in Patients With Acute Myeloid Leukemia (AML) or Myelodysplastic Syndrome (MDS)

Inclusion Criteria: * Diagnosis requirement for phase I patients: * Refractory AML without complete remission (CR) after induction therapy (primary induction failure) or relapsed AML after obtaining a CR. * OR High-risk AML (by ELN criteria; See Appendix C) in complete remission (CR) and has either refused hematopoietic stem cell transplantation OR is currently not eligible for hematopoietic stem cell transplantation OR for whom hematopoietic stem cell transplantation is being reserved for later relapse. This is inclusive of patients with minimal residual disease evidenced by cytogenetics, molecular testing, and/or flow cytometry. * OR Myelodysplastic syndrome (MDS) with excess blasts (\>5%) and progressive disease at any time after initiation of DNA hypomethylator treatment during the past 2 years, OR failure to achieve complete or partial response or hematological improvement (see section 12.4) after at least six cycles of azacytidine or four cycles of decitabine administered during the past 2 years, OR intolerance to azacytidine or decitabine. MDS patients with isolated 5q- abnormalities that meet these criteria after lenalidomide therapy and DNA hypomethylator therapy are also eligible. * Diagnosis requirement for phase II patients: \*Refractory AML without CR after induction therapy (primary induction failure) or relapsed AML after obtaining a CR. Favorable-risk core binding factor (CBF) mutated AML and acute promyelocytic leukemia (APL) will be excluded. * Diagnosis requirement for pediatric cohort patients: \*Refractory AML without complete remission (CR) after induction therapy (primary induction failure) or relapsed AML after obtaining a CR. * Age requirement for phase I and phase II patients: At least 18 years of age. * Age requirement for pediatric cohort: 2-17 years of age. * Available HLA-haploidentical donor that meets the following criteria: * Related donor (parent, sibling, offspring, or offspring of sibling) * At least 18 years of age * HLA-haploidentical donor/recipient match by at least Class I serologic typing at the A\&B locus. * In general good health, and medically able to tolerate leukapheresis required for harvesting the NK cells for this study. * Negative for hepatitis, HTLV, and HIV on donor viral screen * Not pregnant * Voluntary written consent to participate in this study * Patients with known CNS involvement with AML are eligible provided that they have been treated and CSF is clear for at least 2 weeks prior to enrollment into the study. CNS therapy (chemotherapy or radiation) should continue as medically indicated during the study treatment. * Karnofsky/Lansky performance status ≥ 50 % * Adequate organ function as defined below: * Total bilirubin ≤ 2 mg/dL * AST(SGOT)/ALT(SGPT) ≤ 3.0 x IULN * Creatinine within normal institutional limits OR creatinine clearance ≥ 50 mL/min/1.73 m2 by Cockcroft-Gault Formula (adults) or Schwartz formula (pediatric cohort) * Oxygen saturation ≥90% on room air * Ejection fraction ≥35% * Able to be off corticosteroids and any other immune suppressive medications beginning on Day -3 and continuing until 30 days after the infusion of the CIML NK cells. However, use of low-level corticosteroids is permitted if deemed medically necessary. Low-level corticosteroid use is defined as 10mg or less of prednisone (or equivalent for other steroids) per day. * Women of childbearing potential must have a negative pregnancy test within 28 days prior to study registration. Female and male patients (along with their female partners) must agree to use two forms of acceptable contraception, including one barrier method, during participation in the study and throughout the DLT evaluation period. * Ability to understand and willingness to sign an IRB approved written informed consent document (or that of legally authorized representative, if applicable). Exclusion Criteria: * Relapsed after allogeneic transplantation. * Isolated extramedullary relapse (phase II only). * More than one course of salvage chemotherapy for primary induction failure or AML relapsing after CR1 (phase II only). * Circulating blast count ≥30,000/µL by morphology or flow cytometry (cytoreductive therapies including leukapheresis or hydroxyurea are allowed). * Uncontrolled bacterial or viral infections, or known HIV, Hepatitis B or C infection. * Uncontrolled angina, severe uncontrolled ventricular arrhythmias, or EKG suggestive of acute ischemia or active conduction system abnormalities. * New progressive pulmonary infiltrates on screening chest x-ray or chest CT scan that have not been evaluated with bronchoscopy. Infiltrates attributed to infection must be stable/ improving after 1 week of appropriate therapy (4 weeks for presumed or proven fungal infections). * Known hypersensitivity to one or more of the study agents. * Received any investigational drugs within the 14 days prior to the first dose of fludarabine. * Pregnant and/or breastfeeding.