Jo, you do what you need to do, but please know that you have friends that are here for you. If there is anything I can do to help in some small way let me know.
Okay, here comes the difficult bit, trying to give some advice I hope may help. Take what you will and trash the rest and if I say any thing out of line please know it is with best intentionsin mind. I say this because I know that I've had a bloody easy time of it because I have the easier time of it with father/daughter relationships.
First up, you are not your mum. Many of us have a tendency to judge ourselves harshly and believe others see all our self perceived faults. It may sound like I'm blowing smoke up your arse (please forgive the phrasing, it's neither literal, nor am I smoking at the moment - you've been so instrumental in that), but you are awesome. If you can't see that at the moment, give it time and look again. Ask your friends around you back home or on the boards here and they will tell you the same thing as I have. Don't ever doubt it!
As for the kid. There's no sugar coating it I'm afraid, there is going to be tough times. I will promise you, just as people here promise us, it does get easier. You can't be responsible for how she behaves, only your reaction to it.
I only have a couple of suggestions for now. If you can write down how proud you feel and will always feel of her and you can write down how much of a miracle she is and you can add how much you love her no matter what she says to you, do it and make copies (you know a split second reaction can ruin paper ). Also let her know that you will never do anything to deliberately hurt her, but you will maintain discipline and guess what you are human, we are all imperfect and will sometimes get things wrong.
Don't compare yourself to her dad and don't compare your relationships. She should love him and feel safe with him, no matter what. It sounds like she has a pretty fantastic relationship with your dad as well and he sounds pretty special, it may be worth him telling her how much you love her if he is comfortable to do so.
Sorry this is long winded, but the second piece of advice is to occassionally let her stumble and fall when it comes to homework or whatever else she is dealing with. Let her know when she does, it might suck briefly, but that is not failure, that is experience. The only failure is not trying.
Anyhow I'm rambling. Always feel free to talk with me, but look after you first and foremost.
Big hugs to my third favourite Aussie (come on you have no chance up against Warney and Adam Hills - sorry not going to lie to you). Take care of yourself and once again holler loud if there is anything I can do that may help in any way.