Aldesleukin and Pembrolizumab in Treating Patients With Stage III-IV Melanoma

Inclusion Criteria: * Histologic or cytologic diagnosis of cutaneous melanoma, mucosal melanoma, or melanoma of unknown primary that is considered unresectable (stage III) or metastatic (stage IV) * Be willing and able to provide written informed consent/assent for the trial * Have measurable disease evident on radiographs (preferred) or clinical examination; for this protocol, measurable disease is defined as at least one evaluable tumor that is at least 10 mm in longest dimension * Have a performance status of 0 or 1 on the Eastern Cooperative Oncology Group (ECOG) performance scale * Patients must have a brain magnetic resonance imaging (MRI) or computed tomography (CT) (with and without contrast) that is free of active metastases; metastases that have been treated with radiation or surgical resection, are stable for at least 4 weeks and do not require steroids are eligible * Normal cardiac function; patients who have a history of heart disease, or who are over the age of 50 years must have a normal cardiac stress test within the prior 90 days * Normal lung function; patients who have extensive pulmonary metastases or any chronic pulmonary disease history must have pulmonary function testing demonstrating forced expiratory volume in 1 second (FEV1) and forced vital capacity (FVC) \> 65% of predicted values * Absolute neutrophil count (ANC) \>= 1,500 /mcL * Platelets \>= 100,000 / mcL * Hemoglobin \>= 9 g/dL or \>= 5.6 mmol/L without transfusion or erythropoietin (EPO) dep…