Sapanisertib in Treating Patients With Relapsed and/or Refractory Acute Lymphoblastic Leukemia

Inclusion Criteria: * World Health Organization (WHO)-defined acute lymphoblastic leukemia and either: * Relapsed after achieving remission * Refractory to therapy * Newly diagnosed and ineligible for intensive chemotherapy induction Note: patients with T lineage and B lineage ALL are eligible for this trial; likewise, patients with Philadelphia chromosome positive (Ph+) (as long as they are not candidate for other therapies for Ph+) and Ph- ALL are eligible * Bone marrow blasts of at least 10% * At least 4 weeks away from any previous antineoplastic or investigational agent; patients may receive hydroxyurea or glucocorticoids for suppression of leukocytosis, but these must be stopped at least 24 hours (h) prior to initiation of therapy * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Life expectancy of \> 2 months * Total bilirubin =\< 1.5 x institutional upper limit of normal * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal * Creatinine =\< 1.5 x institutional upper limit of normal * Fasting blood glucose (FBG) \< 130 mg/dL * Hemoglobin A1C (HbA1C) \< 7.0% * Relapse after SCT is allowed but no active graft-versus-host disease (GVHD) as per treating physician; also must not exceed the number of prior induction regimens listed above; SCT does not count as line of therapy * Negative serum pregnancy test result; Note: women of child-bearing potential and men must agree to use 1 highly effective method of contraception and 1 additional effective (barrier) method, at the same time, from the time of signing the informed consent through 90 days (or longer, as mandated by local labeling \[e.g. United States product insert (USPI), Summary of Product Characteristics (SmPC), etc\]) after the last does of study drug; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately; men treated or enrolled on this protocol must also agree to use highly effective barrier contraception prior to the study, for the duration of study participation, and 4 months after completion of MLN0128 (TAK-228) administration * Ability to understand and the willingness to sign a written informed consent document * No prior therapy with mTOR inhibitors except for rapalog treatment as part of graft-versus-host (GVH) prophylaxis or treatment * Human immunodeficiency virus (HIV) infected patients (if HIV positive) * HIV infected individuals are eligible provided they meet all the protocol eligibility criteria in addition to the following: * No history of acquired immune deficiency syndrome (AIDS) defining illness other than a historic CD4+ T-cell nadir \< 200/mm\^3 * Prior to leukemia diagnosis, the HIV disease was uncomplicated as evidenced by: * The CD4+ T-cell counts were generally in excess of 300/mm\^3 * The HIV viral loads were less than 200 copies/ml if on anti-HIV therapy * If the HIV is newly diagnosed or there is no history of using anti-HIV therapy, there are no AIDS defining conditions or other HIV-related symptoms * Zidovudine is not allowed as part of the anti-HIV therapy * Patients with diabetes controlled by diet or medication are allowed on trial; controlled diabetes is defined as FBG \< 130 mg/kL in the context of this study Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy =\< 4 weeks prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier; treatment with glucocorticoids, hydroxyurea, and tyrosine kinase inhibitors is allowed up to 24 hour prior to initiation of therapy * Patients with white blood cell (WBC) \> 30,000 are not eligible to start therapy; however, it is permissible to use glucocorticoids and/or hydroxyurea to diminish peripheral WBC to less than 30,000 provided these agents are stopped at least 24 hours prior to the first dose of MLN0128 (TAK-228) * Patients who are receiving any other investigational agents * Patients with known other active cancers; skin cancers (basal or squamous) are exempted * History of allergic reactions attributed to compounds of similar chemical or biologic composition to MLN0128 (TAK-228) * There are no prohibitions of specific medications on the basis of anticipated drug-drug interactions * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, hypertension, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements; no ischemic myocardial or cerebrovascular event, placement of pacemaker, or pulmonary embolism within six months of receiving first dose of MLN0128 (TAK-228) * Any patient receiving chronic corticosteroid administration prior to study enrollment is ineligible * Baseline prolongation of the rate-corrected QT interval (QTc) \> 480 milliseconds or history of congenital long QT syndrome or Torsades de pointes * Concomitant administration of any proton pump inhibitor (PPI) is not permitted during the study; patients receiving PPI therapy before enrollment must stop using the PPI for 7 days before their first dose of study drugs