How to begin getting my mother placed in a facility?

I'm at the end of my rope. I've actually been there but have managed to hold on somehow. I've taken care of my mother in her house for almost 7 years as her dementia has worsened and worsened. I've not had any help at all and haven't been anywhere without her for over 4 years. I never leave the house because I'm working 4-5 jobs at a time because I took on all of her expenses except rent.

She started becoming very mean years ago and now is simply an awful person. She denies everything she does and calls me names, every day with every single thing she does that is nonsensical or dangerous, or both, the weight in my shoulders gets heavier and heavier. I'm so exhausted and so miserable and so stressed out I can't even think straight enough to do my work without making huge, often expensive, mistakes. I feel utterly hopeless and I am utterly alone, both literally and metaphorically.

I've applied for my state's assistance in placing her four times and they've denied her every time, saying she's not "progressed" enough to qualify. This is a woman who talks to her dead parents, saves food for them, hoards newspapers to give to them, who insists she's just had a conversation with my (useless) younger brother (the one who completely disappeared 4 years ago when I finally told him I needed help with her), who leaves with her clothes crammed in grocery bags to go "home" - her childhood home that's 2700 miles away. I've had to lock up everything and now have to buy 3 different locks for 3 different doors because she's decided my dog is hers and she can do what she wants, which means letting him out near the street with no collar and no leash, something I've never done with him because he's a runner. Once she does one weird thing I know it'll never stop, she will continue doing it over and over. I have no way of protecting him from her other than keeping him locked up in the same bedroom I'm closed up in 24/7.

I've literally sacrificed my whole life in order to take care of her and in return I'm called stupid, a liar, an idiot, a bitch, and more often than not I feel so much pressure inside of me it makes me shake. I've become desperate to get out of here before it kills me. It's already ruined me, being around my mother constantly long-term made my older sister move out when she was 15, my older brother leave the moment he graduated, and made me leave Kentucky in 1989 to move all the way to Arizona. My dad drank to deal with her. I've spent my entire adult life getting as far away from my childhood as possible - I kept house differently, I married differently, I raised my kids differently, I strived to be nothing like my mother, I just wanted to escape all the dysfunction and hoarding and depression and manipulation I grew up surrounded by. And I had succeeded.

The moment I knew I had to be her caregiver, in her house, I felt as if God had picked me up and dropped me right back into that awful childhood. Different state, different house, different age, same dysfunction. I've been deeply unhappy the whole time I've been here but as she gets worse I get worse. I have so much anger towards her for things that happened years ago but I'm also fed up with everything here. She's never thanked me, not once, unless in sarcasm. She's never appreciated anything I've done or taken care of or paid for it solved. Angry landlords, pissed off doctors, bill collectors - they've all become my problem. I was prepared to reapply for state assistance but today as I stood in the backyard, after rescuing my dog again after she let him out without a leash, hearing her spit at me that he's her dog and she'll do whatever she wants with him, I felt something break in me. I don't yet know if it was a good break or another bad one but immediately afterward my first thought was getting her placed. I don't know where/how to begin. I know her check will pay for only the most run down of places, I've tried to avoid that, but I'm desperate. I have to get away from here.