Linkal- are you in the US? If you are, I think a best move is to consult with a local council on elders, social services and, if there is an affordable option, an elder law attorney,
You have seen that it is not feasable to be a full time caregiver to your mother and also take care of yourself and your own goals, like studying for the bar exam. If holding on to this rope means that both of you go over the bridge, then your only option is to let go of that rope.
https://bpdfamily.com/message_board/index.php?topic=65164.0;allTaking care of someone who is mentally ill and who has physical challenges may need professionals involved in their care.
Pook's suggestion for assisted living is along that line. However, in the US, elder care is not so straightforward. There are financial rules to know about. Assisted living is self pay and costs several thousand dollars a month. If a person does not pay, or is a behavior problem- the assisted living center can evict them.
Skilled nursing home care is different however there are still financial considerations. Medicare covers short term rehabilitation, like if someone has broken a bone, or needs continued care for a short term illness. Different Medicare plans have different coverages. If a person needs long term care and can not pay, Medicaid can cover costs but there are rules for qualifying for that. Medicaid looks back 5+ years (depending on state) to determine if a parent has gifted their assets, and this will result in a wait penalty.
Do not sign any forms when enrolling your mother. Apparently she is considered legally competent according to the responses you have gotten. She herself needs to voluntarily go, and sign herself. She may refuse. You also need to know what constitutes neglect and what are your legal obligations (this varies by state).
So it may not be possible for you to get her into a facility right away but it will help you to know the rules, what the different facilities are and how to plan for Medicaid, and have a plan in place for yourself. Your mother may agree to get the process for Medicaid going as another option that some states offer is to cover some hours of home health care. She may be intersted in that.
I wanted to be proactive with my BPD mother but she was not cooperative. Also it's not just with BPD. No elderly person wants to go to a home- they want their independence, but sometimes a crisis happens and so it helps to have the roadmap decided if possible. Sometimes the situation is that they need to go to the hospital and then be discharged home. The first person they will look at to care for her is you. That is where you can step in and say you can not take care of her, that she has greater needs. They will need to make other arrangements for her, like temporary skilled nursing.
My BPD mother kept her health and finances secret and remained in her home, refusing assisted living even though she needed that help. Her crisis was financial. She had run out of money and second mortgaged her house, gotten herself into financial difficulty. Her only option was to sell the house, pay the loan back, and use the rest for assisted living.
It may not be possible to plan for moving your mother to a facility- but taking steps to find out how to do that and having a plan will help you with seeing the situation as hopeless. The law protects the elder's autonomy. As you saw, agencies can not intervene without proof of mental incompetence. For my mother, this would have to have involved testing, and medical testifying in court and it would be a waste of time and money, as it would be with your mother.
Your being proactive and learning the various aspects of elder care is being proactive and protecting yourself while also navigating a way to arrange care for your mother if and when that becomes the next step for her.