Peppermint oil can be helpful for pain, but in birds most sources say that it should not be used in birds due to toxicity. Diluting it a lot might be acceptable, but then the pain relief might also be diminished. Most links say to avoid it altogether in birds. Being close to the nostrils, I...
Your vet could be of help with pain management if needed, but in a mild beak injury, the pain might not last too long hopefully. They can use meloxicam or something like Rimadyl for pain.
Do you have a picture of the beak? Yes, those injuries can be very painful for a couple of days. Feeding some mushy wet chicken feed, soft scrambled egg may tempt her to eat. As long as she is drinking, she should eat soon.