The whole of Misty is in key Eb. That's clear from the key signature - it stays at 3♭ all through. What does happen is modulation - as occurs in so many songs, including many classic ones - and indeed, classical music.
By thinking it 'changes' key, you're not going to improvise, or jazz your way through this, or many other standards. It's probably better to consider two or three bars at a time, where you'll inevitably discover ii-V-I going on. With that information, you can play through the changes, making the whole thing sound more cohesive. True, you'll be changing the set of notes (scales, if you like) more often that if yyou just 'changed key', but it's not going to work just 'changing key' for a middle eight, when several of those changes are only vaguely related to that 'change of key' - too many avoid notes are avaiable!