Belinostat and Bortezomib in Treating Patients With Relapsed or Refractory Acute Leukemia or Myelodysplastic Syndrome

Inclusion * Relapsed or refractory acute leukemia * acute myeloid leukemia (AML) other than APL * acute lymphocytic leukemia (ALL) * acute leukemia that has evolved from a prior myelodysplastic syndrome - no requirement for prior therapy * myelodysplastic Syndrome (MDS) - International Prognostic Scoring System (IPSS) intermediate-2 or greater * chronic myelogenous leukemia with myeloid or lymphoid blast crisis * WBC =\< 50 x 10\^9/L; hydroxyurea or leukopheresis may be used prior starting treatment * Prior allogeneic stem cell transplant is allowed provided that \>/= 12 months have elapsed since allogeneic transplant; no graft versus host disease is present; not currently on immunosuppressive therapy * AST, ALT =\< 2.5 x upper limit of normal (ULN) * Female subject who is post-menopausal or surgically sterilized or willing to use an acceptable method of birth control (i.e., oral or injectable hormonal methods; barrier methods such as intra-uterine device, diaphragm with spermicide, condom with spermicide, or abstinence) for the duration of the study * Male subject agrees to use an acceptable method for contraception for the duration of the study * Serum total bilirubin =\< 1.5 x upper limit of normal * Serum potassium \>= 3.5 mEq/L and serum magnesium \>= 1.7 mEq/dL (electrolytes may be corrected with supplementation) * ECOG Performance Status (PS) =\<2 * Creatinine =\< 1.5 x upper limit of normal or calculated or actual creatinine clearance \> 45 mL/min Exclusion * Willing and medically suitable for remission induction with other agents in anticipation of a potentially curative allogeneic bone marrow transplant * Known CNS malignant disease * Prior severe allergic reactions to bortezomib, mannitol, boron, belinostat or compounds of the hydroxamate class or arginine * Grade 1 with pain or Grade \>= 2 peripheral neuropathy or paresthesias within 14 days before enrollment * History of sustained ventricular tachycardia, ventricular fibrillation, Torsade de Pointes, or resuscitated cardiac arrest • History of resuscitated cardiac arrest. Note: persons without pre-existing cardiovascular comorbidities who have experienced resuscitated cardiac arrest in the setting of sepsis ARE eligible provided they have no residual cardiac abnormalities and providing they do not require ongoing medication to manage cardiac issues as an outcome of such an event. * Conduction abnormality or concomitant treatment with an anti-arrhythmic agent to prevent or control arrhythmia * Known congenital long QT syndrome * Clinically significant infection including infection with HIV, or active hepatitis B or C * Significant cardiovascular disease, hypertrophic cardiomegaly or restrictive cardiomyopathy, myocardial infarction within the past 6 months, unstable angina * Baseline QTc interval \> 450 msec * Planned or ongoing treatment with any drug that may be risk of causing Torsades de Pointes * Persistent blood pressure (BP) of \>=160/95 * Serious medical or psychiatric illness likely to interfere with patient participation * Pregnant or nursing * 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, an in situ malignancy, or low risk prostate cancer after curative therapy * Planned ongoing treatment with other drugs thought to potentially adversely interact with belinostat * Strong or moderate CYP3A4 inhibitors * Patient has received other investigational drugs within 14 days before enrollment * If steroids for cancer control have been used, patients must be off these agents for \>/= 1 week before starting treatment. Exception: maintenance therapy for non-malignant disease with prednisone or steroid equivalent dose \< 10 mg/day is permitted.