Social Services would still assess OPs mum for whether she needs to be in a care home before they started paying for it. But since at that point OPs mum would already be in a care home (if OP were to organise this) when her money ran out and as she has dementia, they're not going to decide to pull her out of there and put her in homeless accommodation while they wait for a council flat to become free, just so that they can return to giving her the cheaper option of 4x carer visits per day. Because that would be the alternative accommodation situation and it wouldn't be in her best interests.
OP if you do decide to use your POA in this way before giving it up, all you need to do is contact care homes in your mum's area, visit some, pick one you like and that has a vacancy, and arrange with them for her to move in there. If she has no savings at all, you'd best explain to them about not getting paid until the house is sold, to check that is ok. You inform her social worker it's happening. Beyond checking that what you're doing doesn't constitute an abuse of her (which putting her in a care home where she needs to be, doesn't) there's nothing they can say against it and will have to agree, because it's not them paying so they can't object on those grounds.
[We moved a relative who was in an unsuitable care home (chosen by social services because it was the one closest and with a vacancy at the time) on discharge from hospital, into a nicer one where relative wasn't upset all the time. Social Services had no choice but o agree because relatives needs were being met and that was all that matters. My mum did something similar for my nan, moving her to a care home closer to herself so she could visit more often. Nan was previously in a care home in the area she lived, which was some distance away from my mum. You seem to think it's all down to social services and don't seem to realise the power you have.]
Then sell the house as quickly as you legally can (someone mentioned upthread about there being some legal issues to sort out what with you being both POA for your mum and co-owner of the house), at auction maybe, to release your mum's share of funds as well as your own share. I imagine care homes will be used to this type of situation where the money is tied up in an asset. Or maybe you do the house sale part first and when a buyer is found then your mum moves out to the care home? You'll have to get legal advice on it.