I have been through all of this with my MIL a few years back, and through the work I do I've supported numerous families in the same process. My MIL lacked capacity, and my DH & his sister were awarded POA before capacity was lost.
I have also read the entire thread fully - you've been given some great advice in places, but some of it less so.
There are multiple issues in the OP and because the factors overlap in places, the waters have become extremely muddied - you absolutely need to separate out each point individually, to work out what the issues are, what the consequences are, and how you proceed.
First off, if you are saying you mum has money for around two years care, that must mean she is self-funding her care, and must have undergone a financial assessment from social services as part of the assessments that are carried out before someone can permanently move to a care home. Therefore, the value of her property will have been decided & recorded already, and will have been used to determine whether or not she has to pay for some, all, or any of her care.
At this point, your sister was not living in the house, nor does she fit into the other criteria (as outlined by PPs) to have the property value disregarded anyway. In other words, this ship has likely sailed, and I can’t see how she can expect to move in & hope to have the value of the house disregarded at the discretion of the social services / council, because I cannot see how / why another financial assessment would be carried out until such time all your mum's savings and then assets had been drawn down to £23,250, which is the current threshold where the council will step in to help pay the fees. Here, another assessment would be done so that the council could verify the need for them to pay for her care.
But while mum is self-funding the whole of her care fees, the issue becomes one of whether or not your mum can continue to afford to pay the care home bill, and nothing else. No one can or is going to force you to sell the house, nor can or will they force you to rent it out. The only obligation is for the care home to have their bill paid, and no one cares how or who does that, so long as it’s paid for, using money that has not been acquired illegally.
As POA, if you decide that you are not going to sell the house, and then your mum runs out of savings, it’s up to you to decide how the care home fees are paid, and that may (note: may) mean you can ask the council to pay the care home fees & put a charge for that amount onto the property. However, as others have said, once your mum dies, there isn’t much time in which to pay back that amount – 90 days from death, bearing in mind that interest will be payable on this loan (“the charge”) too.
In respect of your sister moving in and living in your mum’s home rent-free, this is not a deprivation of assets, in the same way that selling the house now and banking the money is not either – in both cases the “asset” is making no money (except for any interest gained while in a bank account of course).
However, if your sister (or anyone else for that matter) moved in, paid rent, your mum paid tax on the rent, and then the rental income was spent on anything that was not required for your mum’s needs, this would almost certainly be considered a deprivation if social services had the need to investigate your mum’s finances. But as before, that’s only going to happen when all the savings & assets that have already been taken into account have been depleted to £23,250 and another financial assessment takes place.
Now, you now have to unpick the complexities of allowing your sister to move into your mum’s house, and to pay for renovations on a home that is not hers. How is all that going to work both from the point of view of your sister claiming in the future that she has put X amount of time & money into the home, either when your mum runs out of savings, or dies? Because given that the house will have already been taken into account as an asset, you will have to draw down against the value of it one way or another.
Assuming that your have to sell the house in order to draw down against it, whether or not your sister could argue that she is entitled to receive the value of the money she spent on renovations is another matter, but given that social services will already have valued it for Y, it will only go up in value if work is done to it, so even when it sells for Z it is only the original value that has to be adhered to, leaving money over that won’t have to be taken into account for care home fees.
All of this is much further complicated by what happens after your mum’s death. You must be aware that your POA ceases the second your mum dies, and from thereon in it’s the executors (assuming your mum wrote a will and appointed executors) who take control. Do you know if she did this? If so, has she shown you a copy of her most recent will? If so, who gets what and who are the executors? Because if you aren’t an executor, you’re going to have to explain to whoever is how and why your sister came to live in the property.
To summarise, your mum has savings and a house, and is in a care home – no matter what path anyone in that situation takes, it is very, very rare for assets to be disregarded, and your sister is stupid for thinking she of all people is the one who has found a loop-hole around of it…it just does not happen that way. The one and only time I knew of a property being disregarded was in the case of someone who only owned half of it (because she'd owned it as a tenant-in-common with her late husband, and his half had been left to their sons), and her half was only worth around £40,000. This -along with the fact that it could technically be impossible to sell the house should the sons choose not to- allowed social services to consider her half to have no value, and was therefore disregarded. But as before, it's a very rare thing to hear of.
As for whether or not you should allow her to move in, and furthermore allow her to spend her money doing the place up, that falls squarely on your shoulders, and no one on Mumsnet can tell you what to do…you just have to be prepared to justify your choices to anyone who may be authorised to ask you for such information, and you have to be prepared to get your sister out the house if / when it has to be sold, or if / when you find out it’s been left to someone else.
Were it me in your situation, and taking into account the fact you say the property is in no fit state to be rented out, I’d be putting in on the market tomorrow, taking whatever figure it goes for, and banking the money in your mum’s bank account. I wouldn’t want any worry or responsibility for what could or couldn’t be held liable for above the bare minimum. But I’m not you. Good luck.