Child's lies and behaviour tearing the family apart

I am at a complete loss, and if something doesn't change soon I will have no choice but to walk away. I have 3 kids, 3, 9 and 10. Miss 10 is a typical kid, can be a bit of a brat sometimes but is generally good. Mr 3 has special needs and is the kindest, sweetest little boy I've ever met. Then there's miss 9. She enjoys annoying myself and her sister, just pushing and pushing until we lose it. Any attempt to discipline or ignore her fails because she will just completely ignore me and keep going no matter the consequences, and if she doesn't get a rise out of us she just keeps going until she does, even if it takes all day. Even when we try to walk away she follows and keeps going. Then when someone eventually does snap she claims to be the victim. For example if 9 pushes 10 past her limit and 10 screams at her, pushes her etc she will come running to me or whoever else is there crying that 10 said something mean or pushed her as if she did nothing wrong. She also lies a lot, trashes the house, breaks everyone's stuff, completely disrespects me and so on.
I had assumed it was an attention issue, so I tried to give her more positive attention, more one on one time, and not give attention to the negative behaviours, but she seemed to think that it meant she could keep getting away with it. I tried time outs, taking away devices, rewards for good behaviour, and everything else I could think of, but it didn't work. I ended up taking her to see a psychologist to try and figure out why she is acting out and how to address the behaviours. They were unable to give me any answers and all it did was make things worse. She told so many lies that they had safety concerns which resulted in a social worker coming to my house every week. Then she started telling the social worker that things are really bad at school and nothing was being done about it, that 10 and I are always mean to her and a whole bunch of other stuff. From there it's just completely escalated.
Last year I would get calls from the school every few weeks telling me she forgot her lunch, which we later found she was leaving in her bag or just deliberately leaving at home. This year I had been checking almost every day that she had lunch, and since the phone calls stopped I assumed the problem was resolved. Turns out she was throwing her food in the bin almost every day and telling her new teacher that I don't feed her, and the teacher was worried about her getting into trouble if the school called me so teacher was making her lunches at school instead. I eventually got a call this week saying I need to bring lunch and found out this was happening. While I was there I mentioned that we had just found out her glasses had been missing for a few weeks (she only needs them at school) only to be told they disappeared in week 2 and that she said they were getting repaired and implied I was too poor to pay for the repairs. 9 also told the teacher that our house burnt down so we were homeless, and a bunch of other stuff that had her believing that she was being neglected and living in poverty. Then when I confronted her about it she told me her teacher is lying. While I found a solution to the lunch situation and have taken her to get new glasses, I just don't know how to make the lies stop.
And now to top it all off the social worker said right in front of her that if she keeps trashing the house and refusing to clean up her mess then I need to stop trying to make her clean it up and do it myself, so of course when I tell 9 to do something like clean her room she tells me to do it myself, she doesn't want to, she doesn't have to. I can't live like this. I'm so tired and miserable all the time, and I just can't do it.