Targeted Therapy in Treating Patients With Relapsed or Refractory Acute Lymphoblastic Leukemia or Acute Myelogenous Leukemia

Inclusion Criteria: * Participants \>= 18 years of age with relapsed/refractory leukemia with a confirmed diagnosis of acute myelogenous leukemia (AML) or acute lymphoblastic leukemia (ALL) who meet the following criteria: * Individuals aged 18-64 years with salvage treatment failures only - defined as relapsed or refractory to after least 1 cycle of salvage therapy * Given the clinical activity and use of hypomethylating agents in AML patients, initial and salvage therapy may include hypomethylating agents * Age \>= 65 years: Refractory to induction chemotherapy - defined as no response to initial therapy or have relapsed after initial therapy * Individuals aged \>= 65 years, with chronic myelomonocytic leukemia (CMML) or myelodysplasia (MDS) that transform to acute leukemia while actively receiving hypomethylating agents (i.e., decitabine or azacytidine) will be considered induction failures and are thus eligible for this trial; for Philadelphia positive (Ph+) ALL, initial therapy and salvage therapy may include steroids and imatinib or dasatinib * Primary patient samples must show in vitro kinase inhibitor sensitivity as determined by the Oregon Health and Science University (OHSU) functional kinase inhibitor screen; for OHSU patients, functional kinase inhibitor screening may be performed as part of this study or through enrollment in eIRB4422 if the identical Food and Drug Administration (FDA)/Clinical Laboratory Improvement Act (CLIA) approved assay is u…