I would do the POA first, and quickly. She needs to do it whilst she has capacity to understand what she is signing. I'm not a lawyer but its sounds as if she would, plus if your wider family agree you can do it yourself, getting three of them, plus a witness, to sign.
Its sounds like your mum (warning, internet is obviously not the place to diagnose) is three or four years less advanced than my mum, though in part she was able to cope prior to her fall because she had a routine, plus we discovered she was not coping as well as we thought. Receiving Attendance Allowance would suggest she does not have capacity, and I am not sure you would get it now anyway, so leave it till after the POA.
The problem is that moving requires learning new routines, and it would take her at least a couple of months to settle. However learning new routines will be easier sooner rather than later. Its sort of like you have to invest some of the current capacity/independence in order to maximise future capacity/independence. Not an easy one, if she is happy and relatively safe.
POA simply gives you a cheque book and the ability to do things on your mums behalf. It is supposed to be a gradual thing so even if one exists, you only use your powers if she can't. My mum had been giving away her bank details to phone callers, so it was important she did not have access to them. Banks are in strange territory regarding capacity as it is not binary. Their immediate instinct was to not allow her access at all, but we reached a compromise of having something similar to a teenagers account, eg a small savings account with no cheque book or direct debits, but with a debit card. I discovered then that she can't use a cash point, but there is a small balance on the account so if emergency money were needed someone could withdraw on her behalf and I could top up. What I do is leave cash in the safe where her medicine kept, and the carer gives her some money every few days to cover her daily trip to Tesco.
Writing this reminds me how lucky I am that she is in very sheltered accomodation. Warden, laundry, a weekly clean, maintenance organised, cooked lunch available, a reception to control visitors, company in the form of a coffee lounge and social events. When you are working with someone with no memory, care and money have been hard enough to organise long distance. There is a lot of problem solving in the early stages, which if everything goes well can help maintain a decent quality of life. Sadder now as my mother is losing her enjoyment of company ond conversation, and I can't really help her much.