You can't force your mum, nor should you try, but is she rational enough to have it explained to her that the financial LPA has nothing to do with her care or where she lives and also that she cannot be forced into a care home, or to have any care she doesn't want, whilst she is deemed to have capacity? Even if she is still against POA it might reassure her a bit, assuming she believes you.
If she still refuses there's nothing you can do (and I hear applying to Court of Protection for deputyship is taking almost a year at the moment).
I bang on about the below a lot, so please excuse me anyone who has seen me mention this before:
For the financial LPA, be very careful when deciding which box to tick in the section about when the POA comes into force. If the donor ticks "only when I lose capacity" you may get into difficulties if the person wants help before they lose capacity. When I registered my mum's LPA with her bank, they assumed she had lost capacity because she had ticked that box and so blocked her access to her account, including cancelling her debit card and cheque book. They gave me access, but she needed her debit card so we had to reverse the whole thing.
We were stuck in a hideous limbo for ages of her being unable to operate the
account (too deaf to pass phone security, refused to let us register her for internet banking or grant 3rd party access), demanding we (I) help her but needing her own debit card.
Also get one or two certified copies of the financial LPA. Any solicitor can do this, though charges vary so shop around. The donor can do it themself but it involves writing some text, dating and signing on every page of every copy.