This pilot clinical trial studies infusion of laboratory-grown donor cord blood cells following combination chemotherapy in treating younger patients with acute myeloid leukemia that has returned or that does not respond to treatment. Chemotherapy drugs work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Chemotherapy also kills healthy infection-fighting cells, increasing the risk of infection. The infusion of laboratory-grown cord blood cells may be able to replace blood-forming cells that were destroyed by chemotherapy. This may decrease the risk of infection following chemotherapy, and allow for more chemotherapy to be given so that more cancer cells are killed.