How do you deal with potentially putting your parent in a home?

I'm 20 years old, and this is the first time I've had to deal with something like this. Ever since I was about 12 I've helped care for my mother. She hurt her knee, it never was fully recovered. She was mobile and all, but she still required my help from time to time with daily things such as cooking, shopping, etc. Two months ago she had a severe stroke, she has made a lot of improvement due to rehab, but still requires things such as transfers to her wheelchair, bathroom, bed, etc. She came home with us (I am currently staying with older sister.) And quite honestly it's been hell, I barely get any sleep due to the fact I have to wake up multiple times in the middle of the night to either change her, or medicate her because she is in some pain, it usually results in me sitting up with her 30+ minutes a time till she goes back to sleep. The only times I get any rest are if my sister has enough time from work + her kids to take over for a while. Even then, she is not strong enough to lift my mother on her own, so I still have to help in those cases.

I believe, and so does the doctor, that the stroke has effected her in ways that most of the time I don't even recognize her as my mother.
Problems we are currently having involve;
-Her wanting to eat all the time, she is overweight but not obese, still, for some reason she's hungry around every hour or two and is picky about her food. If we don't comply she tries to guilt us into giving her a snack or something by trying to convince us she is "starving" or "tired of being hungry all the time." Ontop of the fact she doesn't want to follow the doctors advice on what to eat, and wants to eat junk foods a lot of the time.
-Her wanting to go out all the time. Almost every single day she wants to go out somewhere, or if she hears we're going to go on a shopping trip for about an hour she wants to go. This is a problem because transferring her in and out of our current car is an extreme hassle.
-She is very stubborn and impatient as well, so if we want to go on long trips (say a few hours shopping at the mall for the holidays) within around 40+ minutes she gets very antsy and restless and becomes rather rude towards us, at that point our shopping trips turn miserable and we just end up going home.
-She's been having episodes that are on the borderline dementia, if not there already, where she's completely irrational and thinks, also attempts, to do things she simply can't do such as walk on her own, or get out of bed and put herself into her wheelchair. With her current physical state this is not at all possible in any way, we try explaining to her but she doesn't listen.

At this point, it has become too much even for the two of us. I love her very much, and I realize a lot of her mental problems she's having aren't her fault but they are driving me nuts, on top of the physical stress (I am getting very sore all over due to the multiple transfers everyday, plus no sleep) I honestly feel I cannot do it much longer. We are considering multiple options, including my brother renting a house to stay with her and hire a nurse while he's at work, as well as me going over there every so often to help out.
However setting that up, if we do, could take a few months in which case we are considering putting her in a home until then.

Knowing her though, the fact she hates anything that isn't her home, this would be extremely hard on her, and the thought of that is painful. I feel like all this is killing me, so I know I can't do it much longer, but emotionally I'm feeling worse at the thought of having to put her in a home, even if it's just for a while.

What exactly can I do to get through this? I try to do things such as remember all the times she is, quite honestly, being a bitch to us so I won't care, but that doesn't help much. It's just replaced with memories of the times she's nice and pleasant to be around. I don't know what to do.