Firstly speak to the local authority and see if they are willing to increase the amount allowed for the room.
If the answer is, as I suspect, no you can decide to enter into a top up agreement with the home and pay the difference between what the council are willing to pay and the homes fee.
Otherwise the local authority will have to find a home which is more affordable in terms of the funding they are willing to provide and Ada's financial position.
I am afraid it is a tough time to be both a council and a nursing home. Councils are seeing government funding to be cut to zero in the next few years and nursing homes are going out of business at an alarming rate.