Jack said "Don’t peek", but how could she respond "I'm not"? I thought “I don’t" is right.
Rose means to say, "I'm not (peeking)," in reference to this specific occasion. If she were to say "I don't," it would sound unnatural as it refers to the fact that she does not peek in general, and not this particular time that she is referring to. Thus, "i'm not" works best in this context. Ex: A: Are you going to the mall today? B: I'm not. Ex: A: Don't cheat on the test! B: I'm not (cheating)!