It may be time to rat on your sister. Hove using a debit card without a third party mandate in place will almost certainly be illegal, whilst using your mums assets to meet her own needs is stealing from a vulnerable persons, so a good chance oc a prison sentance.
Research and then decide whether to issue your DSis with an ultimatum or seek help from Social Services.
In Social Services speak to someone who knows about elderly protection, and see if they will help with the Court of Protection process. They will be worried about deprivation of assets, and them eventually having to pick up the bill. They will have lawyers, but might then request the guardianship.
I would not tell the bank as they will simply shut down access to her money. The Court of Protection has a claim to be the slowest moving and least user friendly bit of the public sector. Access would then take months and you would need to use a lawyer and explain what your sister has been up to.
You cannot go on picking up the pieces. Its like modern slavery. (I have a brother who has left it all to me but at least I have POA and control the money.)
So either she agrees to the home, or you go to social services.
Has your mum got a formal diagnosis of dementia? We got my mum to sign the forms, witnessed by a sympathetic priest who knew her well, shortly before we sought a diagnosis. (She had a fall and we needed to step in quickly.) You dont need a lawyer to do a POA, but your mum will need to sign and you need someone to witness confirming she knew what she was signing. Court of Protection could take a year or more (I cannot overstate how slow they are, even for simple transactions and you almost certainly will need legal help) leaving her with no money in the interim.
You might phone the Alzheimer's Society anonymously for their take or try the legal section of their Talking Point Forum.
My instinct would be to approach SS. Not least they can provide interim funding.