I really hear you on this one, I have a 9 going on 13 year old daughter too and she also has phases of being horrible to me/stropping about the place, although they are varied with being absolutely delightful and you don't always know which you are going to get.
There's a couple of things to say. I wouldn't worry too much about her dad and his problems being passed on, if you head over to the teenagers section of MN you will find lots and lots of parents in a similar situation, it's pretty normal for at least some stroppiness to manifest, many from lovely families in which there are not big problems or massive dysfunction- I have seen myself how some of my lovely unproblematic friends had terrible teenage siblings, all from the same household so it is not necessarily connected with that (it can be of course that is a reaction to a specific problem).
The other thing that stands out is that she is almost definitely in pre-puberty and hormones will be going wild. The average age of onset of menstruation is about 12 1/2 which hasn't changed that much over the past decades, but the onset of prepuberty (hormones breast growth, hair growth and general stroppiness) is much earlier, up to two/three years earlier, so by 10 she may well be into this timeframe as will many of her friends as well. If you talk with the other mums you may find they are all saying the same.
Also, you are a single mum and that's sooooo hard, because all that stroppiness isn't directed at two of you, but only one of you- you are essentially the container for all those emotions/stress of being a 10 year old and it's very stressful for you. I am on my own a lot (husband works away) and although I know it's not the same, I still find just doing all the emotional work on my own really exhausting. So, you do need to look after you during this time, go out with friends, have a friend to moan to, treat yourself so that you aren't too battered by it all.
As for what to do, I wish I had a magic solution or really knew what worked, but I am a bit clueless myself. I have done a really light form of love-bombing and it works incredibly well, recently I had a day out with each of my children, just doing what they wanted (all this French and affirmations is what you want to do and unless you need to do them for academic reasons, I might not think this worth battling over) and it really does help bonding, you can have a laugh, reconnect. It doesn't mean they will then be perfectly behaved, but it does make you feel better about them, and help them remember you are a human being with feelings who needs to be treated nicely too.