Recombinant Interleukin-15 in Treating Patients With Advanced Melanoma, Kidney Cancer, Non-small Cell Lung Cancer, or Squamous Cell Head and Neck Cancer

Inclusion Criteria: * Patients must have histological or cytological confirmed malignancy in the following disease groups: melanoma, non-small cell lung carcinoma, renal cell carcinoma or squamous cell head and neck carcinoma, for which no standard effective or curative options are available * Measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 * Documented evidence of disease progression during 6 month period prior to the time of enrollment * Prior therapy requirements: * At least \>= 1 prior completed chemotherapy regimen including chemotherapy, biologic, immunologic or targeted therapy * At least 4 weeks from last dose of prior chemotherapy with resolution of the acute toxic effects of the therapy * At least 2 weeks from completion of prior radiation therapy * At least 4 weeks from last dose of prior investigational therapy * Not receiving any current anti-cancer therapy * At least 4 weeks from last dose of interferon or IL-2 therapy * At least 8 weeks from completion of antibody therapy with anti-checkpoint antibodies, such as anti-cytotoxic T-lymphocyte-associated protein 4 (CTLA4) and anti-programmed cell death 1 (PD1) * At least 4 weeks from last dose of prior other biologic agents * Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 (Karnofsky \> 70%) * Absolute lymphocytes \> 500/mcL * Absolute neutrophil count \> 1,000/mcL * Platelets \> 100,000/mcL * Total bilirubin within normal institutional limits * Prothrombin time (PT)/partial thromboplastin time (PTT) \< 1.5 x upper limit of normal (ULN) * Hemoglobin (Hgb) \> 9 g/dL * Alkaline phosphatase =\< 2.5 x ULN * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) \< 2 x institutional upper limit of normal * Serum creatinine \< 1.5 x ULN or creatinine clearance \> 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * No known central nervous system (CNS) metastases or neurological symptoms possibly related to active CNS metastasis * Females of childbearing potential must have a negative pregnancy test within 48 hours prior to initiation of protocol therapy; NOTE: subjects are considered not of child bearing potential if they are surgically sterile, they have undergone a hysterectomy, bilateral tubal ligation, or bilateral oophorectomy or they are postmenopausal; menopause is the age associated with complete cessation of menstrual cycles, menses, and implies the loss of reproductive potential; by a practical definition, it assumes menopause after 1 year without menses with an appropriate clinical profile at the appropriate age; women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) from the time the consent is signed and for the duration of study participation; 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; women of child-bearing potential and men treated or enrolled on this protocol must also agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) 4 months after completion of rhIL15 * Ability to understand and the willingness to sign a written informed consent document * No history of any hematopoietic malignancy * No active (as defined by requiring immunosuppressive therapy) or history of clinically significant autoimmune disease (as defined by previously requiring immunosuppressive therapy) * No evidence of a clinically significant active infection * No systemic or inhaled corticosteroids within 7 days prior to initiation of protocol therapy; NOTE: use of topical corticosteroids and/or eye drops containing glucocorticosteroids is acceptable * No immunosuppressive therapy within 30 days prior to initiation of protocol therapy * No history of severe asthma, as defined by prior or current use of systemic corticosteroids for disease control, with the exception of physiological replacement doses of cortisone acetate or equivalent, as defined by a dose of 10 mg or less; NOTE: history of mild asthma not requiring daily therapy is eligible * No history of pulmonary disease such as emphysema or chronic obstructive pulmonary disease (COPD), (forced expiratory volume in one second \[FEV1\] \> 2L or \>= 50% of predicted for height and age); pulmonary function tests (PFTs) are required in patients with significant pulmonary or smoking history * No history of human immunodeficiency virus (HIV), active or chronic hepatitis B, hepatitis C or human T-cell lymphotropic virus (HTLV-I) infection; NOTE: a positive hepatitis B serology indicative of previous immunization (i.e., hepatitis B surface antibody \[HBsAb\] positive and hepatitis B core antibody \[HBcAb\] negative), or a fully resolved acute hepatitis B virus (HBV) infection is not an exclusion criterion * Females of childbearing potential and males must be willing to use an effective method of contraception (hormonal, barrier method of birth control or abstinence) from the time the consent is signed, during the duration of study participation and 4 months after discontinuation of protocol therapy * Females must not be breastfeeding * No evidence of clinically significant congestive heart failure, (ejection fraction of 45% or greater) * No platelet or blood transfusions within two weeks of obtaining baseline laboratory values * No blood modifiers while enrolled in the study (i.e., growth factors such as erythropoiesis-stimulating agent \[ESA\] or filgrastim \[G-CSF\]); NOTE: blood transfusions are allowed per institutional guidelines Exclusion Criteria: * Patients who have had chemotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C), or radiotherapy within 2 weeks prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier * Class II or greater congestive heart failure as described in the New York Heart Association Functional Classification criteria * Patients with thyroid disease should be excluded unless their T4 is normal or they are on replacement therapy * Patients with primary brain cancer or known brain metastases should be excluded from this clinical trial * Patients who have received prior anti-CTLA4 or anti-PD1 therapy less than 8 weeks prior to enrollment * Patients who have received prior biologic agents less than 4 weeks prior to enrollment * Patients who have received prior interferon or IL-2 therapy less than 4 weeks prior to enrollment * ECOG score greater than 1 (Karnofsky \< 70%) * HIV-positive patients * Positive hepatitis C serology * Patients who are receiving any other investigational agents * Inability to home monitor blood pressure * 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