This phase I/II trial studies the side effects and best dose of 6,8-bis(benzylthio)octanoic acid (CPI-613) when given together with docetaxel and to see how well they work in treating patients with stage IIIB or IV non-small cell lung cancer. Drugs used in chemotherapy, such as CPI-613 and docetaxel, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading.