BPDMom our family had been in the very same position last year when our D was 16. We had tried it all: PHP, IOP, inhome therapists, therapeutic HS, and after many sleepless nights her dad and I recognized residential as being the only viable option. We knew we had exhausted all other possibilities and her safety was our #1 concern.
You are right no parent / child "wants" residential. However, there is a big difference between needs and wants. Our D needed placement as she wasn't getting relief from any other service.
The one thing that helped me convince her was having a heart to heart with her. We went out for a car ride and I let her know that she DESERVES to live a happy / peaceful life. We could get thru this as a family and would be with her as much as possible. She wasn't being abandoned, shipped away as a punishment. Thankfully we were able to secure space in a program within our state and were only a 45 min. drive away. All residents do need to show that they want help in order to be accepted as there is such a shortage of space /beds with no room for those who aren't ready for the help.
After visiting half a dozen programs, most of which were undesirable

best to us all, we came across the "right house". The staff not only connected with parent, but also with our daughter

She opened up with them during our initial visit and let down her guard just a little. Honestly both my D and I cried ourselves to sleep the first night.
You know your D better than anyone. What do you think will motivate her? During residential, we started talking about a family vacation. It helped motivate her to continue to work hard, take care of herself and gave her something to look forward to.