What should I do if I feel that my mother should stay in a nursing home, but she is resistant to the idea?

It's normal for an elder to not want to go to a nursing home. They have visions of the old style (and there are still too many of these around). If she truly needs a nursing home, then look around for the best one you can find. Then get some help from someone who understands this need (this usually works better than family members). Expect some depression, but see her as often as you can. Spend time with her and encourage her to take part in social activities. She could get to like it. Much depends on the home - and the elder.
Carol