SS will not take her,
I work with a very very vile aggressive nasty stroppy teenage girl, she is 14 and I've been seeing her for six months.
She is horrible, argues with everything just to argue, and is very nasty and spiteful.
She tells her parents that she hates them and wishes they were dead on a daily basis. She has even tried to poison them by putting things in the food (including pouring bleach into a carton of milk, which is why I am seeing her)
She tells people that she is abused, and lies compulsively and she is violent.
SS won't touch her, even when she was making claims of abuse.
Her parents begged SS to take her for a short period, but they won't.
Unless a teen is in serious danger SS never get involved.
I have known of teens that have been dropped off at hospitals and SS offices with a packed bag and told that they are being abandoned, and still then SS won't take them into care, they will just work with the family and try to help.
IIWY I would get a friend (who DD doesn't know) to come round and pretend to be a social worker and tell her exactly the way things are (I doubt you would be able to get an actual SW to come, but you could try)
The reality of it is that even if SS did get involved they would most likely only be interested in helping find a placement with a family member, but if they did agree to take her there is no way she would get a placement with foster parents.
She would be out into a group home or residential facility, she would be sharing a room with several other teenagers, she would have strict rules, that she would have to follow, she wouldn't be allowed to just go out when she wanted or watch tv whenever she felt like it.
She would have no money to buy anything, and no way to get any, so if she needed something she would have to request it, and would only get the most basic/cheap things.
If she didn't follow the rules she could be thrown out and either put in a worse, secure facility, or be given up on and put in a homeless refuge, probably a bedsit.
I have worked with teens in this situation and it is grim, if anything it usually makes them worse and they usually have very little support.
Do you have a family member that your dd can go and stay with for a few days. Preferably someone that will show her exactly how good she has it at home.