Read other letters about this article
Good for you for being concerned about this and prepared to intervene. Knowing where to start is the hardest part.
Advice along the lines of "you have to have her committed, right now" is great, and probably what should happen in this case, but unfortunately, the system is not set up that way. The gatekeepers of the services that protect people from themselves are the professionals in mental health and law enforcement, not the family members. This system is designed to protect people from the abuse of being locked up at anyone's whim, but it unfortunately doesn't always give credibility to those who are experiencing life with a troubled person, and who know them best.
So recognize that this is a process, that will involve gathering information and evidence over time, and ensuring that the gatekeepers have all of that information in front of them as often and for as long as it takes for them to make the right decision. The first time she's admitted, as one letter writer noted, she can be on her best behaviour and be discharged, or discharge herself against medical advice once the 72 hours is up. The first time she's admitted, they'll be prepared to give her the benefit of the doubt. But this doesn't mean you shouldn't try taking this first step. The next time, or the next, or maybe the seventh time this happens, the "system" and the gatekeepers will begin to see what you see, and your chances of finding a solution will get better.
Not easy. Probably not fair. But it's what you and your sister have to work with, knowing that working with your mother is just not possible right now.
PS - I have such limited patience with those who are prepared to spend effort on telling someone how stupid they are when they're here looking for help. What exactly do you think you're contributing to the world? If only life were as simple as you'd like it to be.