I'm guessing here so forgive me if I've got things wrong. If your mum has been in her council house for 65 years, presumably she can buy it at a considerable discount under the Right to Buy scheme. Assuming also she has enough money to buy it. That would mean when she dies you would inherit the (paid for) house rather than it being returned to council housing stock now she no longer needs it.
As LPA your role is to act in your mum's best interests and also, you're not supposed to personally benefit from managing her money. I'm not sure if this would be considered a benefit to you.
I think it may well depend on what savings she has and who is funding her care. If she has very little and the council are funding, the house isn't an asset at present because it is rented. But if she has enough to buy her house I'm guessing she must be self-funding? If you were to do the house purchase (on her behalf) it would then be an asset and would presumably be worth far more than the discounted rate your mum would have paid. This would mean that if her savings ran out, you would need to sell the house to pay for her care! I'm pretty sure that's not what you have in mind!
If you were thinking you could spend her savings on the house and that if she had no funds in the bank, SS would have to take over her funding, I'm afraid you are wrong, for the reasons above.
But without knowing more detail, it's a tricky situation! For example, if she has £150,000 sitting in the bank and could buy the house at a discounted price of, say, £50,000, that would still leave £100,000 to pay for her care. You might even be able to argue that it was definitely in her best interests because if her cash funds ran out (i.e. her care costs came to more than she had in the bank) she could then sell the house to pay for care. But if she has only £50,000 in the bank and you spend the lot on buying the house, then ask SS to fund her, that would certainly be seen as a deliberate deprivation of assets and they (the council) would just make you sell the house and keep paying.
As others have said, the council may not agree to a sale if your mum is unlikely to return to the property so you'd need to check that. A lot to think about! Your mum is 96 so realistically is unlikely to live past 100. So you need to calculate the cost of about 4 years of care and then do your sums. If it is possible to buy and your mum has enough money to still fund her care costs, it would certainly be a worthwhile thing to do. (In my humble opinion that is!)