This phase I trial studies the side effects and best dose of transducer of regulated CREB activity 1/2 (TORC1/2) inhibitor INK128 when given together with epidermal growth factor receptor (EGFR) inhibitor AZD9291in treating patients with advanced EGFR mutation positive non-small cell lung cancer that has spread to other places in the body (advanced) and has progressed after treatment with an EGFR tyrosine kinase inhibitor. TORC1/2 inhibitor INK128 and EGFR inhibitor AZD9291 may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth.