Temsirolimus, Dexamethasone, Mitoxantrone Hydrochloride, Vincristine Sulfate, and Pegaspargase in Treating Young Patients With Relapsed Acute Lymphoblastic Leukemia or Non-Hodgkin Lymphoma

Inclusion Criteria: * Diagnosis: * Patients must have second (2nd) or greater relapse of pre-B ALL, T-cell ALL, lymphoblastic lymphoma, or peripheral T-cell lymphoma; patients may not have refractory disease * Patients with leukemia must have had histologic verification of the malignancy at the most recent relapse, including immunophenotyping to confirm diagnosis * Disease Status: * Leukemia: patients with leukemia must have an M3 marrow with or without extramedullary site of relapse OR an M2 bone marrow with an extramedullary site of relapse; patients with central nervous system (CNS) 3 status are not eligible for enrollment * Lymphoma: patients with non-Hodgkin lymphoma must have either measurable or evaluable disease * Patient's current disease state must be one for which there is no known curative therapy or therapy proven to prolong survival with an acceptable quality of life * Karnofsky \>= 50% for patients \> 16 years of age and Lansky \>= 50 for patients =\< 16 years of age; patients who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score * Prior Therapy: * Patients must have fully recovered from the acute toxic effects of all prior anti-cancer chemotherapy, defined as resolution of all such toxicities to =\< grade 2 or per the inclusion/exclusion criteria * Myelosuppressive chemotherapy: * Patients with leukemia or lymphoma who relapse while receiving standard maintenance chemotherapy with steroid, vincristine pulses and oral outpatient chemotherapy will not be required to have a waiting period before enrollment onto this study * Patients who relapse while they are not receiving standard maintenance therapy, must have fully recovered from all acute toxic effects of prior therapy; at least 14 days must have elapsed after the completion of cytotoxic therapy, with the exception of hydroxyurea * Note: cytoreduction with hydroxyurea in patients can be initiated and continued for up to 24 hours prior to the start of protocol therapy * Hematopoietic growth factors: at least 14 days after the last dose of a long-acting growth factor (e.g. Neulasta) or 7 days for short-acting growth factor; for agents that have known adverse events occurring beyond 7 days after administration, this period must be extended beyond the time during which adverse events are known to occur; the duration of this interval must be discussed with the study chair * Biologic (anti-neoplastic agent): at least 7 days after the last dose of a biologic agent; for agents that have known adverse events occurring beyond 7 days after administration, this period must be extended beyond the time during which adverse events are known to occur; the duration of this interval must be discussed with the study chair * Immunotherapy: at least 42 days after the completion of any type of immunotherapy, e.g. tumor vaccines * Monoclonal antibodies: at least 3 half-lives of the antibody after the last dose of a monoclonal antibody with the exception of blinatumomab; patients must have been off blinatumomab infusion for at least 7 days and all drug-related toxicity must have resolved to grade 1 or lower as outline in the inclusion and exclusion criteria * Radiation therapy (XRT): at least 14 days after local palliative XRT (small port); at least 84 days must have elapsed if prior total body irradiation (TBI), craniospinal XRT or if \>= 50% radiation of pelvis; at least 42 days must have elapsed if other substantial bone marrow (BM) radiation * Stem cell infusion: no evidence of active graft versus (vs.) host disease and at least 84 days must have elapsed after transplant or stem cell infusion * Study specific limitations on prior therapy: patient may not have received prior therapy with an mTOR inhibitor * Note: intrathecal (IT) methotrexate (MTX) that is given up to 72 hours prior to initiation of systemic chemotherapy per ADVL1114 counts as protocol therapy and not prior anti-cancer therapy; IT MTX given \> 72 hours prior does not count as protocol therapy * Platelet count \>= 20,000/mm\^3 (may receive platelet transfusions) to initiate therapy * Patients must not be known to be refractory to red cell or platelet transfusion * Creatinine clearance or radioisotope glomerular filtration rate (GFR) \>= 70ml/min/1.73 m\^2 or a serum creatinine based on age/gender as follows: * 0.6 mg/dL (1 to \< 2 years of age) * 0.8 mg/dL (2 to \< 6 years of age) * 1.0 mg/dL (6 to \< 10 years of age) * 1.2 mg/dL (10 to \< 13 years of age) * 1.5 mg/dL (male) or 1.4 mg/dL (female) (13 to \< 16 years of age) * 1.7 mg/dL (male) or 1.4 mg/dL (female) (\>= 16 years of age) * Bilirubin (sum of conjugated + unconjugated) =\< 1.5 x upper limit of normal (ULN) for age * Serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 225 U/L; for the purpose of this study, the ULN for SGPT is 45 U/L * Gamma-glutamyl transpeptidase (GGT) =\< ULN for age * Serum albumin \>= 2 g/dL * Shortening fraction of \>= 27% by echocardiogram, or ejection fraction of \>= 50% by gated radionuclide study * Pulse oximetry \> 94% on room air * Baseline chest x-ray; patients with active infectious disease or pneumonitis are not eligible * Serum triglyceride level =\< 300 mg/dL and serum cholesterol level =\< 300 mg/dL * Random or fasting blood glucose within the upper normal limits for age; if the initial blood glucose is a random sample that is above the upper normal limits, then a follow-up fasting blood glucose can be obtained and must be within the upper normal limits for age * All patients and/or their parents or legally authorized representatives must sign a written informed consent; assent, when appropriate, will be obtained according to institutional guidelines Exclusion Criteria: * Pregnant or breast-feeding women will not be entered on this study; pregnancy tests must be obtained in girls who are post-menarchal; males or females of reproductive potential may not participate unless they have agreed to use an effective contraceptive method * Patients receiving stable or decreasing doses of corticosteroids for =\< 7 days prior to enrollment, or who are receiving increasing doses of corticosteroids, are not eligible for enrollment; the exception to this is pulsed steroids used for maintenance chemotherapy * Patients who are currently receiving another investigational drug are not eligible * Patients who are currently receiving other anti-cancer agents are not eligible (except patients receiving hydroxyurea, which may be continued until 24 hours prior to start of protocol therapy) * Patients who are receiving cyclosporine, tacrolimus or other agents to prevent graft-versus-host disease post bone marrow transplant are not eligible for this trial * Patients who cannot receive asparaginase are not permitted on trial; substitution with Asparaginase Erwinia Chrysanthemi is acceptable * Cumulative prior anthracycline exposure must not exceed 400 mg/m\^2 (each 10 mg/m\^2 of idarubicin or mitoxantrone should be calculated as the isotoxic equivalent of 30 mg/m\^2 of daunorubicin or doxorubicin) * Patients who are currently receiving therapeutic anticoagulants (including aspirin, low molecular weight heparin, and others) are not eligible * Patients who are currently receiving angiotensin-converting enzyme (ACE) inhibitors are not eligible due to the development of angioneurotic edema-type reactions in some subjects who received concurrent treatment with temsirolimus + ACE inhibitors * Enzyme-Inducing anti-convulsants: patients who are currently receiving enzyme-inducing anti-convulsants (i.e., phenytoin, phenobarbital, or carbamazepine) are not eligible * Patients with CNS 3 status at enrollment are not eligible * Patients must have no pre-existing grade 1 or higher ulcerations, fistulas, mucosal lesions, or skin barrier breakdown * Patients who have an uncontrolled infection are not eligible * Patients with known optic nerve and/or retinal involvement (because it may not be possible to safely delay irradiation) are not eligible; patients presenting with visual disturbances by history or physical exam should have an ophthalmological exam and magnetic resonance imaging (MRI) within 14 days prior to enrollment to determine whether there is optic nerve or retinal involvement * Patients with known Down syndrome, Fanconi anemia, Kostmann syndrome, Shwachman syndrome or any other known bone marrow failure syndrome are not eligible * Patients who have received a prior solid organ transplantation are not eligible * Patients who in the opinion of the investigator may not be able to comply with the safety monitoring requirements of the study are not eligible