Hi megabee,
from what it looks like, there seems to be 2 problems here:
1. she is overtired and therefore keeps waking up at night. 11pm is just tooooo late for a 2 year old to sleep. They need AT LEAST 10 hours night sleep at this stage. 9pm is a more reasonable bedtime. Once she sleeps earlier, she will wake less at night.
2. she does not seems to know how to fall asleep herself, or go back to sleep on her own. You need to train her on this. Try pat her back, rather than giving her milk when she wakes at night. Hope it helps.