Need to get Grandma into nursing home. Where do I start?

Start calling nursing homes and come right out and say exactly what needs to be said: "My grandma is going to need a NH soon, what do I need to do to get started?" Tell them everything, including the legal stuff about POA and her unwillingness to discuss it. I went through this a few years ago for my MIL (who also was adamantly refusing to leave her apartment) and the social services workers I spoke to at various NHs were all very understanding... they've heard all this before and are used to dealing with anxious family members who don't know the drill.

Once you know the basics of how to 'get started' it makes it a little less daunting. Thankfully in the case of my MIL, we didn't need a NH, but I learned a lot.

How do you get her to the doctor? Maybe you have to "trick" her, for her own good. Get her ready for an outing and, oh my, we're at the doctor. Don't even say the word doctor, appointment, etc. "Someone" needs to meet you? Some authority figure says she has to go? You'll have to come up with something to get her there since it's very important that she get evaluated, etc.

Maybe call some local nursing homes and ask them for help in getting themselves a new patient?

Maybe they can help you get her on medicaid? I'm sure the paperwork is challenging and they must have experience dealing with those kind of difficulties.

Good luck!

Your mother certainly did have all the legal standing required not to move your grandmother into her house. So, your mother must have agreed to this. Does your mother also now agree that your grandmother needs to be in a Nursing Home? - because you won't get very far if she doesn't.

If your grandmother's agitation is marked, seize the moment and call an ambulance. Once you're at the hospital, you should be able to take it from there - ask for help and information.

Meanwhile, have a browse around this site: https://www.aging.pa.gov/aging-services/caregiver-support/Pages/default.aspx

First step might be to get her to the Dr. so she can be evaluated. Is she decisional? If yes there is not much you can do. With the exception...next time she falls, and there will be a next time call emergency services (911) and she will be transported to the hospital. IF she is admitted prior to her discharge state that she is not safe at home and that you can not safely care for her. That might get her into rehab. This is only if she is admitted. And unfortunately many are not admitted but placed "under observation"
Your other option would be to contact a lawyer well versed in Elder Law. You would or your Mom would have to get Guardianship. Once you have Guardianship you can make decisions for her for health and finances. This is not an easy job and there will be costs. The costs will come from Grandma not you or your mom.
The other thing that you should look into would be the possibility of application for Medicaid if her assets will not cover Assisted Living or Memory Care. (and to that if she is diagnosed with any form of dementia the move to memory care should be done rather than moving her again as she declines further.)
Look for a facility that will accept Medicaid so that she will not have to be moved as her assets dwindle.