How do we convince my mother-in-law she needs to go to a nursing home?

It's likely that if she moved in with you, you'd quickly find yourself doing the lion's share of work. My hubby, at one time, wanted his father to move in with us. We still had 2 teenage daughters at home. They love Grandpa, but said they'd leave if he came.
Years later my hubby understands how stressful and difficult that would have been. At the time he was all for me staying home and running my life around his father. As it ended, we got in home care for him (they were lousy, sadly) and I wound up going out to his place 2-3 times a day to care for him. It was a better choice than having him move in.
If your MIL can possibly afford some in home caregivers, try that route. Be aware that once she begins the constant falling---her options are going to be very limited. And the "compliance" when she's in the NH is totally normal. My mother will sing and dance for anybody who is coming to her house to give her therapy, then she sits down and doesn't move for days.
Your hubby's guilt is normal---but help him be strong.

Moving in with mil or vice versa WILL NOT solve any problems. She will still fall, she will still not eat, she will still sit and not exercise. You guys will not be able to change any of that just because you live under the same roof. Besides not solving any problems, it will create a whole new list of them and your husband's relationship with his Mom will change completely. I speak from experience with my Aunt, who is living with me. I can't force her to do anything and she's in MY house. I suspect if I was in HER house it would be worse. You guys are in a hard place. Please recognize you are not doing her any favors by moving in.