How to get someone into nursing home from assisted living? This process seems ridiculous, is there an easier way?

Grandmother has been in assisted living in Florida for several years. She cannot transfer, shower or dress herself. She cannot walk and has Alzheimer's. Somehow they have kept her at assisted living. She has now depleted her funds and the children are supplementing the cost of $6,000.00 a month. They want to put her in a nursing home, but she has been on a wait list in Florida for over 3 years. They keep saying she is number one on the list, but she never gets a spot. Now the family plans to send her to NY and get her into a nursing home here. A nurse told them to get grandmother here, take her to the hospital with "altered mental status" and have them admit her for 3 days so she can then be discharged to nursing home. To me, this seems ridiculous. Is there an easier way? Is the process different in Florida?