We have whining issues, too. The amazing part is how quickly she can shut it down. Whining, tearing up, then tell her she can't have it until she asks in her nice voice.
"May I have some milk, please?"
It's hard to keep from busting out laughing.
The key to shutting down whining is not to reward it. Ever.
E-v-e-r!
Random reinforcement is actually more reinforcing that reliable reinforcement.