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

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

What this study is about