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

* 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:

* 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

* 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

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

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 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 must have no pre-existing grade 1 or higher ulcerations, fistulas, mucosal lesions, or skin barrier breakdown

What this study is about