I would also say you should get some legal advise on this as this can be tricky and to stop any worry or stress of not knowing. I was in a similar situation to you.
Even though not married the ex could have a claim on being able to live in the house as the mother with care on the behalf of keeping a roof over your daughters head, so that bit is right. This would stop when your daughter is 16 or 18 can't recall which and the house would go back to you and your ex would have to leave and you could "evict her" then. You would have a nice investment nest egg then!!
As you were not married what was yours before you lived together is now back to yours, what was hers is now hers and anything you paid for between you in your time together you would decide who has what or buys the others share. I could not be bothered to argue over kettles etc
This also goes for property and from what you have said she had her own flat before she moved in with you so the house was your property before she moved in with you so would go back to being yours. The only thing for any financial interest in the property, she would need to show that she has contributed to the property financialy through paying towards the mortgage which you say she has not. Key though, is it in your name only? etc. Also if she can show she has paid for big repairs or major decorating.
How nice is her flat compared to the house. Do you think by doing the following she would move back to it. One thing you can do, (but please think very long and very hard about this before doing it as you have very good contact with your daughter and that is priceless and this could upset things), is to decide yourself if she would have a reasonable financial claim on the house. If you think she would not or are 50/50 then make out you will put it up for sale, just get a few estate agents round to value it should do the trick and start the process. She will then go and see a solicitor who will then advise her on how strong her claim is. If they think she has a good case you will soon get a letter telling you to take it of sale and then a court case would follow. Beware that even if they don't think she has a good case you may get a letter with some waffle in it but nothing firm as they will still try it on. She may decide to move out if told she does not have a claim.
As basszebra says def get some advice on it before you do what I have said above (that's what I was advised to do but in the end she moved out before it came to it when she found out she could get a nicer place on benefits!!!). Might cost you a couple hundred pounds as I found the first half hour free thing not much good. Also be very careful not to put your contact you have at risk especially if you have overnights as it is soul destroying and to be avoided and you may regret it. Once they start being dificult with contact it is [censored] with cancelling at last minute, sudden illness, excusses you name it. In my case she made it clear from day one that she thinks the children are hers alone and I am to be excluded with minimal contact but you seem to be in a better place.