I don't know if she is bipolar, just sounds like she is troubled. The circumstances surrounding her life, with no emotional support, probably contributes to her problems. If she lies about things, and contradicts herself means she has some personality issues not necessarily bipolar. I could never put up w/a person who lies constantly about things, they would be dropped like a bad habit. Everyone has problems but there is no room for that kind of lying. You need trust, and it's difficult to have in a relationship like that.
There are too many single women out there worth more of your time than her. Stop hanging around just cuz she looks nice to you if she's driving you mad. It's not worth it, beauty fades and you will be stuck there putting up with nonsense.