This phase II trial studies the side effects and how well giving busulfan and etoposide followed by peripheral blood stem cell transplant (PBSCT) and low-dose aldesleukin works in treating patients with acute myeloid leukemia (AML). Drugs used in chemotherapy, such as busulfan and etoposide, work in different ways to stop the growth of cancer cells, either by killing the cells or by stopping them from dividing. A PBSCT may be able to replace blood-forming cells that were destroyed by chemotherapy. This may allow more chemotherapy to be given so that more cancer cells are killed. Aldesleukin may stimulate the white blood cells to kill cancer cells. Giving busulfan and etoposide together followed by PBSCT and aldesleukin may be an effective treatment for AML.