Unless your Dad has signed as a guarantor for the fees, then the LA can only take her half - that is assuming that she goes first.
The situation as I understand it is that M & D own the property as joint tenants. That means that on the death of the survivor the other one inherits automatically, irrespective of what the Will says. Thus if mum dies first, the LA will only be able to claim back against her share of the property. However, if Dad goes first Mum will inherit the lot and thus the LA can claim back all of it.
As for "splitting" the ownership, I assume you mean severing the joint tenancy to make it a tenancy in common. This means that each person owns a distinct share of the property. When one dies, their half does not automatically pass to the other by survivorship, it passes under the terms of the Will. You say that Dad cannot now "split" it because of mum's mental state. That is incorrect. Dad can still sever the tenancy. All he needs to do is serve a notice on mum saying he is severing the joint tenancy.
As for what the LA can and can't do, I suggest you look at CRAG -
http://www.dh.gov.uk/...alasset/dh_115533.pdf
Mazie - might be a good idea just to review your wills make sure they don't require any changes?