In terms of what you all need for her tobreturn home, yes she needs an urgent MH assessment (whether she meets the criteria for an urgent referral, I do not know. Our local children's MH services team still treat up to 18 I believe, but I don't know if that is for chileen who have been referred before they're 16. One way or another, need the ball rolling on this, and with the difficulties you have getting to see a doctor, it would be good if this process could be begun while not at home, and someone else got her to appointments. (I got my dd to appointments but wasn't allowed in for them anyway, so don't feel you are letting her down in any way if someone else is doing that.) If she gets a standard referral she may have to wait months for an assessment, which could be at an inconvenient time and place, so in your position I would be asking for transport (a driver) to takebyou to such appointments - and your own medical appointments too, if that is possible. (I have no idea if it is but whould that be useful practical support?)
Next, you mention respite. How much do you need, when and how often and how near. Write it all down. They need to provide transport too, of course.
I've always been told that SW will do all they can to keep a family together and for DCs to be at home, so if you are showing you want her back (with vital support) and that it's best for you all, then wouldn't it be better for them to provide the support? Of course, that makes a lot of work for them, but that is their job. The reason they get paid out of taxes is precisely to help and support families who have these needs! :)