Not sure what to do about my grandmother's care. Should she stay in nursing home after "rehab" time is up?

I am a 30 year old only child of two only children. My parents have both passed away as well as my other grandparents. My 88 year old paternal grandmother is my only remaining family and I am hers, so her care has fallen to me. Since this August she has had 3 back to back UTIs. She has been in the hospitals for falls with utis 4 times in the past 12 months. After the most recent episode in which her neighbors had to come get me because they could hear her inside her house but she couldn't get to the door. I went and she had her phone right beside her but wouldn't call for help (was also an issue when she had a medical alert button she wouldn't push it when needed). The dr at the hospital does not think she can safely return home. They sent her to the nursing home for skilled care. With the most recent UTI she had major incontinence issues and massive confusion. Usually once she is on antibiotics for a few days the confusion improves, but this time it hasn't. The PT at the nursing home says she has no safety awareness and if she starts to fall she just falls and doesn't try to steady or catch herself. We don't have a close relationship to begin with, and she is fiercely independent. Today she called me and told me she needed to call a bank in a town she hasn't lived near in 15+ years because she had been robbed and they had taken all of her checkbooks. She never had any checkbooks with her in the facility, but was adamant that someone had taken them. It's frustrating to not have anyone to bounce ideas off of, because no one around me has dealt with this. I'm also exhausted after being my mom's primary caregiver during her final months of stage 4 cancer and am still dealing with selling my parent's homes and belongings. Any advice appreciated.