I must say, even with your update, I'm afraid I see things differently. I experienced all the same things you have with my mother (an alcoholic and abusive behavior to me) from the time I was a child up until most recently. She will be 95 years old this month and is now in a nursing home.
I had a sister who was the leach, and money I gave my mother, ended up in the pocket of my sister all too often. Being an alcoholic for so many years is a very difficult addiction to beat. Especially if they are a mean drunk, as they often attack the people who are trying to help them.
I would not totally abandon her. Especially before she is able to secure more manageable health insurance. I would suggest your sister and you pay for her home insurance and real estate taxes, so she will not lose her home and any medical need she has until she can obtain medicare, and then any other medical costs she may have that medicare does not cover.
I would tell her and your brother that you are willing to do this only if he gets himself a job, even if it is part time for $150 a week to contribute to the household and pay the cable and food bill. Irregardless if this happens, I would still assume the insurance, taxes and medical co-pays etc. You will most likely need to research available policies in her area, and select the best coverage plan you can find for her, as it sounds like neither her or your brother have the ability to do so. Forget about gratitude. You may never get it.
Your brother was undoubtedly a help for your mother during a time she needed the most help, and you can not discount that. If she is in very poor health, she will most likely continue to need help as she ages. At some point she will need to go into a nursing home, and you will be able to get her regular medicaid, but if there is no surviving spouse, they will take the house.
Another option for you. Florida has what is called a Diversion Program, where they will contribute a portion of money (anywhere from $1000 to $1300 a month towards the cost of an assisted living facility. Also, if she was married to a Veteran for 10 years or more who did not re-marry or does not have another living spouse, she could also receive another $1000 a month towards the assisted living home. The facility will take her social security and the Diversion payment will go directly to the facility.
When my mother could no longer care for herself (and was too abusive to live with) my sister and I got her qualified for regular medicaid and the diversion program. It paid for a less expensive but acceptable place we found, but the monthly rent was about $400 a month short, which my sister and I split and paid each month. We also paid $50 a month into a petty cash account for her each month, and took care of purchasing any other needs she might have (clothing, etc.) She did not own a house, so in your case she would need to sell the house.
They tell you she can not have any funds, but that is not actually true. If both your sister and you split the proceeds of the house between you and set up a care account for your mother, to be used only for the care of your mother, it will pass the mustard. The catch is, both you and your sister will have to pay taxes on the money as if it were income you received. However, in your case this would still be cheaper for you than having to pay out of pocket every month your mother's short fall.
Just some things for you to consider.