What a sad ending, I'm sorry Fidget. Did she have dementia? At least you were with your parents and tried your best with your mother in law. If she didn't want to go, what's to be done... Unfortunately, dementia or not, they can be very difficult and awkward and will not accept help. She should've accepted that stair lift, it would've saved her life in all probability.
No, she didn't have dementia, but she did have a bad case of stubborness. This was caused by her wanting independence in her own home but she sadly wasn't capable of independent living. Independence would have been fine and admirable if she hadn't needed so much care, but it was additional difficulties for us that could have been avoided if she had moved into our house. She could have lived 'with us' or kept her own privacy and council in her own room, which had room for her TV and some of her own furniture, etc.