Sapanisertib in Treating Patients With Stage IV or Recurrent Lung Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed stage IV or recurrent squamous cell lung cancer or KRAS mutant lung cancer that harbors any of the NFE2L2 mutations or KEAP1 mutations; any KEAP1 mutation will be eligible * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm (\>= 2 cm) with conventional techniques or as \>= 10 mm (\>= 1 cm) with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Patients must have completed at least 1 prior line of systemic therapy; patients who have declined first line therapy or for whom first-line therapy would be clinically inappropriate, will be considered eligible for the trial * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Life expectancy of greater than 3 months * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin within normal institutional limits * Fasting serum glucose =\< 130 mg/dL or hemoglobin A1C (HBA1C) \< 7.0% * 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 within normal institutional limits OR creatinine clearance \>= 50 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Patients with controlled diabetes are allowed on study; controlled diabetes is defined as fetal bovine serum (FBS) =\< 130 mg/dL in the context of this study * The effects of MLN0128 (TAK-228) on the developing human fetus are unknown; for this reason women of child-bearing potential and men must agree to practice 1 highly effective method of contraception and 1 additional effective (barrier) method, at the same time, prior to study through 90 days (or longer, as mandated by local labeling \[e.g., United States Package Insert (USPI), Summary of Product Characteristics (SmPC), etc;\]) after the last dose 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; any woman who becomes pregnant while receiving MLN0128 (TAK-228) will be removed from the trial; 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 120 days after completion of MLN0128 (TAK-228) administration; men must agree not to donate sperm during the course of this study or within 120 days after receiving their last dose of study drug * Ability to understand and the willingness to sign a written informed consent document * Ability to swallow oral medications * Known human immunodeficiency virus (HIV) positive patients who meet the following criteria will be considered eligible: * CD4 count \> 350 cells/mm\^3 * Undetectable viral load * Maintained on modern therapeutic regimens utilizing non-CYP-interactive agents Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 2 weeks prior to the planned start of study treatment or those who have not recovered to baseline or less than grade 2 from adverse events from prior treatments * Patients who are receiving any other investigational agents * Patients with untreated central nervous system (CNS) metastases; patients with treated CNS metastases who are off steroids are eligible * History of allergic reactions attributed to compounds of similar chemical or biologic composition to MLN0128 (TAK-228) * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements; no ischemic myocardial or cerebrovascular event, class III or IV heart failure, placement of pacemaker, or pulmonary embolism within six months of receiving first dose of MLN0128 (TAK-228) * Baseline prolongation of the rate-corrected QT interval (QTc) \> 480 milliseconds, or history of congenital long QT syndrome, or torsades de pointes * Pregnant women are excluded from this study because MLN0128 (TAK-228) is an mTOR agent with the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with MLN0128 (TAK-228), breastfeeding should be discontinued if the mother is treated with MLN0128 (TAK-228) * Patients previously treated with an mammalian TOR (mTOR) or PI3K inhibitor * 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 * Uncontrolled diabetes mellitus (fasting plasma glucose \> 130 mg/dL despite optimal medical management of hyperglycemia) * Known hepatitis B surface antigen-positive, or known or suspected active hepatitis C infection * Patients receiving histamine H2 receptor antagonists before enrollment must stop using these medications for at least 24 hours before their first dose of study drug