Crizotinib is a drug that is approved for the treatment of non-small cell lung cancers (NSCLC) that contain an alteration in the ALK gene. In many patients, the response to crizotinib does not last, and the cancer begins to grow (on average, after less than a year on treatment). The purpose of this study is to see if adding the investigational drug STA-9090 can make crizotinib work better at preventing lung cancer from growing. STA-9090 works by blocking heat-shock protein 90 (Hsp90), which cancer cells need to grow and survive. Crizotinib is taken orally (by mouth). STA-9090 is given intravenously (by vein).