Phase I Trial of AZD1775 and Belinostat in Treating Patients With Relapsed or Refractory Myeloid Malignancies or Untreated Acute Myeloid Leukemia

Inclusion Criteria: * Patients must have one of the following, histologically or cytologically confirmed: * Acute myeloid leukemia (AML) \[non- acute promyelocytic leukemia (APL) AML\] * If previously treated: * AML that is relapsed or refractory to at least one prior line of therapy * If previously untreated, must meet all of the following: * \>= 60 years of age * Secondary or therapy-related AML * Does NOT bear favorable cytogenetic and/or molecular features, eg, core-binding factor abnormalities, FLT3 Internal Tandem Duplication (FLT3-ITD) negative/NPM1 mutated, biallelic CCAAT/enhancer binding protein alpha (CEBPA) mutation without FLT3-ITD * Chronic myeloid leukemia blast crisis (CML-BC) * Relapsed or refractory to at least one Bcr-Abl-TKI-containing regimen * Myelodysplastic syndrome (MDS), must meet all of the following: * Higher risk MDS \[intermediate-2 or high risk by the original International Prognostic Scoring System (IPSS)\] * Relapsed, refractory, or intolerant to at least one prior line of therapy containing hypomethylating agents (deoxyribonucleic acid \[DNA\] methyltransferase inhibitors) * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 50%) * Total bilirubin =\< 1.5 × upper limit of normal (ULN) for the laboratory unless resulting from hemolysis * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 × ULN for the laboratory * Creatinine within normal limits for the laboratory OR creatinine clearance \>= 60 mL/min/1.73 m\^2 (estimated glomerular filtration rate \[eGFR\]) for patients with creatinine levels above the ULN for the laboratory * Human immunodeficiency virus (HIV)-infected persons are eligible if they meet other eligibility criteria including the following: * No prior acquired immune deficiency syndrome (AIDS)-defining condition other than cluster of differentiation (CD)4+ cells nadir \< 200/mm\^3 * Pre-leukemia CD4+ cell count \>= 250/mm\^3 * Willing to adhere to antiretroviral therapy regimen with minimal overlapping toxicity and PK interactions with the experimental agents in this study; no zidovudine- and no ritonavir-containing regimens and no 3-drugs-in-1 pill regimens containing pharmacologic boosters are allowed; recommended regimens are integrase inhibitors combined with tenofovir and emtricitabine * Women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and 4 months after completion of AZD1775 and belinostat administration * Ability to swallow medication * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Clinical picture indicative of leukostasis or evidence of disseminated intravascular coagulopathy * Other investigational agent within 3 weeks prior to initiation of study therapy * Ongoing toxicities \>= grade 2 from prior therapy * Acute promyelocytic leukemia (APL, M3) * Active central nervous system (CNS) leukemia * History of allergic reactions attributed to compounds of similar chemical or biologic composition to AZD1775 or belinostat * Stem cell transplant within previous 3 months prior to initiation of study therapy * Major surgical procedures =\< 28 days before beginning study treatment or minor surgical procedures =\< 7 day before beginning study treatment; no waiting required after placement of a vascular access device * Uncontrolled infection * Pregnant or nursing; women of childbearing potential must have a negative serum pregnancy test performed within 7 days prior to the start of study therapy * Note: Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with AZD1775/belinostat * Circulating blast count \>= 50,000/uL within the week preceding enrollment * Current candidacy for a potentially curative allogeneic stem cell transplant, unless declined * Corrected QT (QTc) interval \>= 450 ms (ie, grade 1 or higher) on electrocardiogram (ECG) prior to initiation of study treatment * If baseline QTc on screening ECG is \>= 450 ms (ie, grade 1 or higher): * Check potassium and magnesium serum levels * Correct any identified hypokalemia and/or hypomagnesemia and repeat ECG to confirm QTc interval * For patients with baseline heart rate (HR) \< 60 beats per minute (bpm) or \> 100 bpm, manual measurement of QT interval by cardiologist is required, with Fridericia correction applied to that manual measurement to determine the QTc for eligibility consideration * Note: For patients with HR 60-100 bpm, manual measurement of QTc interval and use of Fridericia calculation is NOT required * Any of the following related to risk of torsades de pointes and sudden cardiac death: * History of sustained ventricular tachycardia (VT), ventricular fibrillation (VF), torsades de pointes, or resuscitated cardiac arrest unless currently addressed with an implanted cardiac defibrillator * Concomitant treatment with an anti-arrhythmic agent to prevent or control arrhythmia; agents used for rate-control of atrial fibrillation are permitted provided that they are not prohibited due to potential drug interactions * Known congenital long QT syndrome * Second degree atrioventricular (AV) block type II, third degree AV block, or ventricular rate \< 50 bpm or \> 120 bpm * Unstable angina, myocardial infarction or New York Heart Association (NYHA) class III/IV congestive heart failure within 30 days preceding study enrollment * Ongoing or planned treatment with any of the following: * Atorvastatin * Strong inhibitors or inducers of cytochrome P450 family 3, subfamily A, polypeptide 4 (CYP3A4); as part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * If any of these agents have been used, patients must be off them for \>= 2 weeks before starting study treatment * Any known UGT1A polymorphism, heterozygous or homozygous * History of prior therapy with belinostat or AZD1775 * Active gastrointestinal (GI) conditions that might predispose to drug intolerance or poor drug absorption * Receiving any other therapies for cancer treatment (with the exception of gonadotropin-releasing hormone \[GnRH\] agonists for prostate cancer); Note: hydroxyurea is allowed before initiation of study treatment and for the first 5 days of study treatment * Diagnosis or treatment for another malignancy within 3 years of enrollment, with the exception of complete resection of basal cell carcinoma or squamous cell carcinoma of the skin, any in situ malignancy, or low-risk prostate cancer after curative therapy * Medical, psychological, or social condition that, in the opinion of the investigator, may increase the patient's risk, interfere with the patient's participation in the study, or hinder evaluation of study results