Yes, when I lose Mum I will inherit anything left in accordance with her Will.
In your friend's case if a good price was paid for the half house and the Mother still had the cash or had spent it (not given it away) that would be Ok, but then the Mother would still have an asset and would be expected to use it to pay her fees. If the rental income along with any other income/savings she has covered the fees, no problem.
If not it becomes a bit difficult, as the daughter did not live in the house as her carer Social Services would fund the care but take a Charge over the property to recoup the money on sale of the property as happened with my Dad's second wife (his widow) when she ran out of money. Again her heirs inherited what remained after the Charge had been settled