I know that she is a child/teenager and her feelings are totally valid
Well, all feelings are valid. Including yours.
OP, I'm sorry you're having such a tough time. What support do you have in place for yourself?
You have escaped dv. That is no small thing, it's a fucking huge achievement and you should be very proud. You've cared for and brought up your DD, you work hard and you deserve to be happy.
Next, while all feelings are valid and need to be heard/expressed/acknowledged, the stories that are behind those feelings may not be true.
I've messed up as a mother.
I've failed.
we are not a real family
My biggest dream in life was to be a good mother and I've failed at that.
she just seems to hate me and tells me she hates her life.
All of these beliefs sound really hurtful and upsetting. Are they true? Are you sure they are true? Because they don't seem to match up what you're saying about all of the things that you've done so far, which would suggest you are an admirable, loving, and excellent mother, even if you are having a tough time right now.
I highly recommend this book:
https://www.goodreads.com/book/show/46674.Feeling_Good
Which is great on learning how to challenge common depressive thought patterns.
And last to your daughter ... it sounds like she is hurt and angry. Lots of big feelings. Lashing out. Yes, some of that is teenagerhood. And some of it may need to be heard and perhaps it might help to see if she needs some counselling/therapy, too? What about family therapy? It sounds like you both could use some extra support, even if you are close, sometimes we need some external input to help take the pressure off. Then you can start to heal your relationship.
The stories your DD is telling herself and you are also worth questioning, examining and testing for truth. As a PP said upthread, listening can also include asking questions (gently), encouraging insight, and coaching to help with skills where a child may need a bit of extra support.
For example - is it true that 'everyone else' is a big happy family? Really?
Active listening is a fantastic skill to learn. It sounds like you're doing well listening, but here is an article that may be useful:
https://www.peacefulparenthappykids.com/read/brilliant-listener
It's going to be okay, OP. You're doing a great job. You'll get there.