Volasertib and Vincristine Sulfate Liposome in Treating Patients With Relapsed or Refractory Acute Lymphoblastic Leukemia

Inclusion Criteria: * Patients must have histologically confirmed diagnosis of Philadelphia-negative ALL by bone marrow biopsy or aspirate * Patients must have \>= 5% blasts in the bone marrow * Patients must have refractory disease, disease relapse or progression after at least two prior systemic chemotherapy or immunotherapy regimens * Note: Exceptions may be made if a patient is deemed unfit for first-line salvage therapy by the treating physician; such cases should be clearly documented * Patients with a history of CNS (central nervous system) leukemia are eligible if they are not symptomatic from current CNS involvement; if there is CNS involvement that is known prior to enrollment or identified subsequently, it will be treated accordingly with intrathecal chemotherapy per the treating physician * Patients must exhibit an Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2 * Patients must have adequate organ function within 14 days prior to registration, as defined below: * Total bilirubin =\< 2 x institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/aspartate aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SPGT\]) =\< 3 x ULN * Creatinine =\< 2 X ULN * Females of child-bearing potential (FOCBP) and males 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 for 6 months following completion of therapy; should a female patient become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately * NOTE: A FOCBP is any woman (regardless of sexual orientation, having undergone a tubal ligation, or remaining celibate by choice) who meets the following criteria: * Has not undergone a hysterectomy or bilateral oophorectomy * Has had menses at any time in the preceding 12 consecutive months (and therefore has not been naturally postmenopausal for \> 12 months) * FOCBP must have a negative pregnancy test within 14 days prior to registration on study * Patients must have the ability to understand and the willingness to sign a written informed consent prior to registration on study Exclusion Criteria: * Patients who have had chemotherapy, immunotherapy, or radiotherapy within 2 weeks prior to entering the study or those who have not recovered from adverse events (=\< grade 1 or patient's baseline) due to agents administered more than 2 weeks earlier are not eligible * Patients may not be receiving any other investigational agents within 7 days of registration * Patients may not be receiving any medications that are known to prolong QT interval unless reviewed and approved by the principal investigator (PI) * Patients who have a history of allergic reactions attributed to compounds of similar chemical or biologic composition to volasertib or VSLI are not eligible * Subject may not have had hematopoietic stem cell transplant (HSCT) meeting any of the following: * Is within 2 months of transplant from cycle 1 day 1 (C1D1) * Has clinically significant graft-versus-host disease requiring treatment * Has \>= grade 2 persistent non-hematological toxicity related to the transplant * Donor lymphocyte infusion (DLI) is not permitted \< 30 days prior to study registration * Patients with \>= grade 2 sensory or motor neuropathy are not eligible * Fridericia's corrected QT (QTcF) prolongation \> 470 ms or QT prolongation deemed clinically relevant by the investigator (e.g., congenital long QT syndrome); the QTcF will be calculated as the mean of the 3 electrocardiograms (ECGs) taken at screening * NOTE: The formula used to calculate QTcF can be physician's choice, but it must be used consistently throughout the study * Patients who have an uncontrolled intercurrent illness including, but not limited to any of the following, are not eligible: * Ongoing or active infection requiring systemic treatment * Symptomatic congestive heart failure (\>= class 3) * Unstable angina pectoris * Cardiac arrhythmia * Psychiatric illness/social situations that would, in the investigator's opinion, limit compliance with study requirements * Known human immunodeficiency virus (HIV) infection * Known John Cunningham virus (JC) virus infection and/or progressive multifocal leukoencephalopathy (PML) * Known clinically active hepatitis A, B, or C infections * NOTE: Patients with chronic hepatitis C virus (HCV) or hepatitis B virus (HBV) infection may enroll if other laboratory criteria are met; those with HBV surface antigen positivity may enroll only if maintained on appropriate suppressive antiviral therapy for the duration of enrollment in the trial * Second malignancy that requires active treatment * Any other illness or condition that the treating investigator feels would interfere with study compliance or would compromise the patient's safety or study endpoints * Female patients who are pregnant or nursing are not eligible