Combination Chemotherapy and Peripheral Blood Stem Cell Transplant Followed By Aldesleukin and Sargramostim in Treating Patients With Inflammatory Stage IIIB or Metastatic Stage IV Breast Cancer

Inclusion Criteria: * Patients with inflammatory (stage IIIb) or responsive stage IV breast cancer with metastasis to soft tissue and/or bone; responsive stage IV disease is defined as patients who achieve a PR (\>= 50% reduction in measurable tumor burden) or CR following initial chemotherapy for metastatic disease or patients with locally recurrent disease (chest wall/axillary nodes) who are rendered disease-free following surgery or radiation therapy without receiving chemotherapy; bone disease is categorized as responsive if there is demonstrated sclerosis of prior lesions with no new lesions * Patients should have received 4-7 cycles of an Adriamycin and/or taxane-based regimen for stage IIIb or stage IV disease; locally recurrent (chest wall/axillary nodes) patients rendered NED by RT or surgery do not need to receive chemotherapy for stage IV disease prior to Cytoxan/Taxol * Patient has received Cytoxan 4 gm/m\^2 x 1 and Taxol 250 mg/m\^2 x 1 per FHCRC protocol 506.03; Cytoxan/Taxol must be given after all other chemotherapy is completed and before transplant * Stem cells were collected after mobilization with Cytoxan/Taxol or after mobilization from an FHCRC approved cytokine protocol; if syngeneic collection, PBSC's were collected by using G-CSF according to FHCRC protocol 753; patient has an adequate number of peripheral blood stem cells stored (\>= 2.5 x 10\^6 CD34+ cells/kg) * The patient must have the capacity to give informed consent; the patient must have signed an approved consent form conforming with federal and institutional guidelines * Hepatic function: Bilirubin =\< 2 mg%; SGOT or SGPT =\< 2.5 x institutional normal * Renal function: Creatinine =\< 2.0 mg/dl or a creatinine clearance \>= 50 mg/min * Pre-Study tests have been performed as outlined in the Study Calendar * Patients will begin IL-2/GM-CSF therapy if they meet the following criteria post-transplant: * Can start therapy 30 to 100 days after transplant * Karnofsky performance status \> 60 * ANC \> 1,000 cells/mm\^3 and platelets \> 30,000/cells/mm\^3 (transfusion independent) for at least 5 days before starting therapy * Total bilirubin =\< 2.5 x upper limit of normal * SGOT =\< 2.5 x upper limit of normal * Creatinine =\< 2.0 mg/dl Exclusion Criteria: * Patients with a Karnofsky Performance Score less than 70 * Patients with a left ventricular ejection fraction less than 50 % (LVEF must be performed in patients with symptoms of CHF, abnormal cardiac exam or history of Adriamycin therapy total dose \> 400 mg/m\^2) * Patient is pregnant * Patient is seropositive for the human immunodeficiency virus * Patients with a history of seizures * Patients with hypersensitivity to E.coli preparations * Patients with active auto-immune disease * Patients with clinically significant pulmonary disease, i.e., diffusion capacity corrected \< 60% of predicted; patients with pulmonary problems should be evaluated with appropriate pulmonary studies and/or consult * Patients with a history of CNS lesion (brain or carcinoid meningitis) * Patients with significant active infection precluding transplant * Patients who have had more than one prior chemotherapy regimen for stage IV disease or a prior transplant for any stage disease * Patients who have had CD34+ selection of their PBSC products * Patients will not receive IL-2/GM-CSF therapy if they: * Are \> 100 days from transplant * Have documented disease progression after transplant * Have an active infection * Manifested cardiac complications during the initial transplant period, including arrhythmias (that required therapy), congestive heart failure, angina, myocardial infarct, or decreased LVEF to \< 45% * Currently have pericardial effusions, pleural effusions or ascites * Manifested pulmonary toxicity during the initial transplant period and have a diffusion capacity corrected =\< 60% * Are on steroids * Currently have a Grade 3 toxicity from BuMelTT * If the patient does not wish to receive the therapy