Hi, I am no expert. However you might Google "deliberate deprivation of assets".
I would assume that if the market value of the house at the point she transferred it is greater than the amount of debt you paid off, the local authority might consider that she had deliberately deprived herself of assets to avoid paying care home fees. There will be a few wrinkles including mortgage interest you have paid since, the fact that sales prices and achievable market value can be quite different, and that your mother is entitled to make gifts out of capital up to, I think, £3,500 each year.
In short, you may be liable but you might need expert help.
One good place to start might be the legal section of the Talking Point forum on the Alzheimers Society. They are a pretty board church so wont fuss about what sort of cognitive impairment she might have.
In the short term though I don't think you need do anything. She is on the Housing Register and told she will be entitled to HB. This will help as this will open the door to accessing suitable social housing. Presumably you want to clear her house and sell it.
Where you may then need to be careful is if the LA want her to contribute towards her care, on the basis that she would have had assets if she had not given them away. They may have a case and you might then need to find some help in arguing the exact value of the assets she has deprived herself of. So carry on as you were, but don't spend the money yet.
And I am more than happy to stand aside if someone with direct experience and greater knowledge comes along.