When I Wake sounds like a Count Basie piece. Try Lil' Darlin'. Casey's Song has a bit of Louis Armstrong's vibe. You might like this recording he did with Duke Ellington. This is just the tip of the iceberg for these artists, obviously.
Other recs based on those two tunes: Miles Davis (particularly his '50s albums like Milestones, Kind of Blue, Round About Midnight and the Workin'/Steamin'/Cookin'/Relaxin' sessions), Bill Evans, Oliver Nelson's Blues and the Abstract Truth album. Other key artists you might enjoy: Art Blakey, Charles Mingus, John Coltrane, Coleman Hawkins, Lester Young. Check out the links in the sidebar for hundreds of other recommendations of songs and albums from every era of jazz's history.
More specific to Whiplash, the movie's music and philosophy were very much based on Buddy Rich, so you might want to check him out. For example: The movie's opening theme sounds like it's based on Willowcrest by Buddy's band. The movie's climax is based on Buddy's famous solos on the West Side Story medley - this one is a good example. And of course, the infamous Buddy Rich bus tapes were probably used as a basis for the JK Simmons character, at least in part.
The reactions to the movie have been very interesting. I hated it, to be honest, and most of the musicians I know in real life agreed. But here in /r/jazz, the majority seems to have enjoyed it and gotten some inspiration from the film. And non-musicians seem to almost universally praise it. Like I said, I didn't dig it, but I'm trying hard not to get too far into the whole "stop liking what I don't like" vibe. If it helps more people like you to discover jazz then it's got to be a good thing, right?