I'm so sorry to hear about your struggles.
My partner has BPD and his parents asked him to leave age 17. Obviously I'm not a parent but I do have some understanding of BPD from a personal and professional stance.
Most importantly, you have to take care of yourself. So hard to do, but you are super important.
I would recommend reading "Stop Walking on Eggshells: Taking Your Life Back When Someone You Care About Has Borderline Personality Disorder" it has amazing guidance for parents, partners etc.
In terms of housing, accepting her back into the home means they will not provide her with any housing. Avoid speaking to housing and try and contact a mental health charity like Mind to get advice.
I would strongly recommend reading the book first just to get some clarity!
I hope I've been some what helpful