I might just say it one more time.
SHE COULD GO TO HER MUM'S HOUSE.
Her mother is happy to have her but there are rules there - the same ones.
So she doesn't want to go there.
People have asked if the OP would be the same if it was her DS instead - well, the DSD's mother IS the same - sounds like everyone is reaching their limit with her.
Forcing her into prostitution, as some have said? Er, no, She could choose to look for a job, and choose to do that while living with either her father or her mother, (a choice of 2 houses, neither of which contains her boyfriend) while also being polite and helping out a bit around the house. She is CHOOSING not to do that.
I suspect that if a deadline of a month was put in place, and if the DSD made a serious effort in that time (stopped being rude, started to help out a bit and made a serious effort to job hunt) then if a job hadn't actually been secured that deadline would be extended. What you want to see is evidence that things are seriously changing.
To those who are saying the OP is harsh - what would you be doing to ensure this wasn't still going on in 10 years time? How long would you give her? She's had help with getting interviews and hasn't turned up. What would your support consist of, and in the meantime, would you expect help around the house from this adult, and how would you get her to give it if she refused? Would you martyr yourself picking up after her and cooking for her while she was rude to you? Especially if her own mother is refusing to pander to her?
Lots of posters have said they couldn't throw her out. Ok, so what would you do that the OP hasn't done? Or are you seriously suggesting that you would be thrilled to run around after an her while she treats you like dirt?