No particular hotel recommendations (we recently stayed in the Hampton Inn Times Square, which was great but a bit above the rate you quoted) but agree that staying in midtown is best - you'd like to walk to Times Square, Central Park, Lincoln Center, Columbus Circle. Everything else is in Manhattan is a short subway ride.
If it's your cup of tea go to the Rock and Roll Hall of Fame Annex in Soho - they have a great (temporary) exhibit on John Lennon's NYC years. Also David Byrne's big suit.