This phase II trial studies how well giving combination chemotherapy and peripheral blood stem cell transplant followed by aldesleukin and sargramostim works in treating patients with inflammatory stage IIIB or metastatic stage IV breast cancer. Drugs used in chemotherapy, such as busulfan, melphalan, and thiotepa, work in different ways to stop the growth of tumor cells, either by killing the cells or by stopping them from dividing. A peripheral stem cell transplant may be able to replace blood-forming cells that were destroyed by chemotherapy. This may allow more chemotherapy to be given so that more tumor cells are killed. Aldesleukin may stimulate the white blood cells to kill breast cancer cells. Giving aldesleukin together with sargramostim may kill more tumor cells