[john70t]

I would suggest a family meeting and put all the cards on the table.
-Nursing homes are extremely pricey.
-Part-time nursing assistance will not be cheap either. You will occasionally need this to have a life of your own sometimes and ensure her care.
-A lot of work will be involved as dementia patients can be mobile and smart.

If the sister wants to have mom live in some sterile nursing home, or with her (requiring home care 24/7), then her irrational control issues.....ummm....I meant 'concerns'......should be discussed in front of the whole family.
Perhaps another POA be appointed, so a solution can be agreed upon.

I'd suggest you all attend local Alzheimer meetings and talk with other people.
Get references.
Find options.
Make schedules.

Also, the house should be safety-proofed:
Stairs/rugs/thresholds and other tripping hazards should be fixed. Glass and other items should be out of reach. Stoves might need switches. Doors might need buzzers so she doesn't walk across town. Toilets and tubs will definitely need handrails.