Aisling, the full time care here differs from the UK model in some ways, one of which is that in UK, if I have it right, there are Care Homes (for people who can't live alone, but don't need nursing care) and nursing homes, for people who's condition is such that they need more care than a Care Home gives. Here, we just have Nursing Homes. They have to have nursing staff on duty, 24 hours a day. Once the person moves to a nursing home, then, as the Manager of William's place put it, it is their HOME until they die. They all provide end of life care - unless there is, in the end, some medical reason the person needs to be in hospital.
You have not failed, Aisling. We've all fought the Dementia War on behalf of our loved ones held captive - but we were never going to win. We all knew that from the start. All we were doing was putting up a damned good fight anyway, because maybe by us fighting, ground won't be lost so fast. And we can't do nothing.
There is no place for guilt in losing a battle that you had no hope of winning. Rather, you can (when you get enough rest, and the dust settles), say that you may have lost - but you went down fighting.