Moving a family member into a nursing hone is traumatic for most of us. The good news is that most people with advanced Alzheimer's do really well in the company
of others with the same condition. Your mother's spirit and behavior are likely to improve.
Thankfully, times have changed and most nursing homes these days are quite decent, with good nutrition and activity programs, but most of us still have the image of the horrors of old, so even if you could reason with your mother, it would be very difficult to convince her. Because of her advanced dementia, talking to her about that definite a move will really aggravate her.
I suggest that you look on your own at homes in your vicinity. After you've taken the tour of a facility you think will work for her, spend a couple of hours just observing. This will give you a better feeling for daily life.
When you find one that feels suitable for her, there are two options for you:
If you think you can do a gradual transition, make arrangements for the two of you to have lunch and ask the staff to pair you with one or two other residents who are outgoing and social. Don't mention to her that you're planning for her to live there. Instead introduce her to your table mates with something complimentary about her. An example: "This is my mother, Sarah. She is a wonderful cook." After a few more visits, you can suggest that she might want to spend more time with her new friends. You can give her the "good news" that they have a really nice room where she can stay for a few days.
However, if this may not be feasible for you and instead you'll have to move her straight in. In that case, talk to the staff about their transition program. She'll need some time to adjust without your presence. Ask staff to let you know when she's ready for you to visit. When you first get there, you may have to tell her a "loving lie," i.e. that you're going on a trip "“ or you need to do major renovation on your house - and you've found a nice "hotel" where she can stay until you return. Keep your tone joyful, as hard as that sounds and bury whatever guilt feelings you may have. It may be hard for you to use "loving lies," but keep in mind that her telling her the truth would be unnecessarily hurtful.
Once you get the go-ahead from the staff, make your first visit short and sweet. Keep the conversation really positive. She'll likely ask you repeatedly when she's going home. Avoid telling her that she's now living there, rather divert her by talking about her new friends, activities, and food (if it's good.) If it gets to the point when you get frustrated and find it too hard to be upbeat, tell her you need to run an errand and will see her later. This first visit can be as short as you need. The fact is she'll most likely have forgotten about your visit a few minutes after you leave. However, if the staff reports that she's agitated after your visit, you might want to give her more time to adjust. She'll settle in and you'll be able to resume regular visits.