Social services won't be remotely interested in a 17yr old.
She's of working age I think? Unless it's been raised to 18? I'm going to assume it hasn't. So she needs to do two things. First claim universal credit and second get to her new GP for health investigation into neurodiversity/mental health.
If she refuses to do either of these you kick her out. There's no other way if you can't afford to feed her or pay the extra in bills. Unwell or not, she has to grow up. She can't spend her life being babied by a mother who is unwell herself.
If she's kicked out and engages with the council she'll be helped to apply for UC, encouraged to see her GP if needed and possibly given temporary accommodation (if there's anything available and she's got the sense not to turn it down) while she bids for a council flat or helped into private rental she wouldn't otherwise have access to. They have schemes that help with the deposit and landlords that will take benefits claimants, for example.
Universal credit will expect her to look for a job. So she either needs signing off sick from the doctor so she's not required to look for work or she needs to look for a job. Having no skills or qualifications means she'll have to get a job that doesn't require any, like others in the same situation. It's not something to panic about it's something to get on with.
She can forget college. Unless you're funding her living expenses while she lives at home, she can't afford it. You can't claim UC as a full time student who isn't looking for or available for work. If you're signed off and claiming for ill health, you can probably do college as a part time student. I'm unsure of the details of the rules. UC isn't a way to fund yourself through college though. Student loans aren't enough to pay for everything that's needed and if she's someone who struggles with life she is unlikely to be able to study and work part time to top up the loans. You also don't sound like you'd be in a financial position to stand as guarantor for her student accommodation costs, which they'll want. So moving out and going to college/university isn't happening.
If she's going to remain living with you while she goes to college then she can come off UC at that time and quit whatever job she's found, attending college instead. September is a long way away so she should still claim UC now.
You can't find her a job OP that's ridiculous. Nobody is going to employ someone who is either too lazy or too mentally disturbed to apply for it themselves!
Agree she needs a bed if she's living there. Unless she's made homeless, the only way she's going to get her own place is by getting a job and paying for it. Majority of landlords won't rent to people on benefits.
If you didn't want her you should have left her where she was, if the ex girlfriend wanted to live alone she'd have either kicked DD out if it was her home or she'd have ended the tenancy if it was a rental, either if which would have made DD homeless and eligible for help from the council.
You instead took her in, so get her a bed (or she can get one out of her UC payment) and she needs somewhere to put it. Either she shares a room with her sister or you turn the living room into her bedroom, depending on the setup of the home and practicalities. If eldest doesn't like it, she can get her own place and move out as at 18 she's an adult and responsible for herself.
Get the cat checked by a vet. Idk much about cats but a known difficult person, a cat crying when you touch it, vomiting food and growling at the difficult person - I'd be wondering if the cat had been physically harmed by her.
Youngest DD getting her own place is going to involve a flat share, lodging or a HMO, she's not going to be able to afford a flat to herself. Be realistic.
Eldest DD may fair better if she has a good job. Otherwise the same goes for her.
It's no good everyone want want wanting. The situation is what it is and you'll all have to get on with it. Using whatever means is available to you.