Mount Sinai is also a good choice, but it, too, is on the upper East Side--it's at 5th Avenue and 101st Street.
St. Luke's is at 113th and Amsterdam on the upper West Side. Apparently its emergency room is now considered one of the best in the city; 25 years ago, when I lived at 117th and Columbus, if I got sick I would take a cab over to Lenox Hill.