Infusion of Expanded Cord Blood Cells in Addition to Single Cord Blood Transplant in Treating Patients With Acute Leukemia, Chronic Myeloid Leukemia, or Myelodysplastic Syndromes

Inclusion Criteria: * Patients 10 to 65 years old with a hematologic malignancy in need of hematopoietic cell transplant who are \> 30 kg and without a suitable related donor * Patient must have hematologic malignancy that meets institutional eligibility requirements for cord blood transplant * Malignancies included are: * Acute leukemia, including acute myeloid leukemia (AML), biphenotypic acute leukemia or mixed-lineage leukemia, acute lymphoblastic leukemia (ALL); all patients must be in complete response (CR) as defined by \< 5% blasts by morphology/flow cytometry in a representative bone marrow sample with adequate cellularity to assess remission status * Myelodysplasia (MDS) International Prognostic Scoring System (IPSS) intermediate (Int)-2 or high risk (i.e., refractory anemia with excess blasts \[RAEB\], refractory anemia with excess blasts in transformation \[RAEBt\]) or refractory anemia with severe pancytopenia or high risk cytogenetics; blasts must be \< 10% in a representative bone marrow aspirate * Chronic myeloid leukemia excluding refractory blast crisis; to be eligible in first chronic phase (CP1) patient must have failed or be intolerant to tyrosine kinase inhibitor therapy * High dose TBI regimen: 10 to =\< 45 years * Intermediate intensity regimen: 10 to =\< 65 years * Patients 10 to =\< 45 years: Lansky (\< 16 years old) or Karnofsky (\>= 16 years old) \>= 70 or Eastern Cooperative Oncology Group (ECOG) 0-1 * Patients \> 45 to =\< 65 years: Karno…