I agree with jessbow - dont act in haste. You will have to plan everything first.
Care homes have a duty to make sure that someone in their care who is removed is going to a safe place. They will not let you just take her (even with POA) until you have a a care plan and equipment to meet her needs. You mentioned that she had been in hospital, so (depending on why she was admitted and what happened) her needs may have changed since before she went in. You will need to find out what these needs are and then put in place a care package which will meet them.
The big thing is - who is going to pay for it? If your mum is going to need help with funding then the maximum that SS will usually offer is 4 carer slots (usually half an hour long) per day and do not usually provide any night time care. If she needs care during the night so that your/your dads sleep is interrupted then an awake night carer is eye-wateringly expensive. There usually comes a time with people with dementia when their needs become so high that even if you are there indefinitely, 24/7 and you have 4 carer slots a day, plus maybe find someone who would sit with your mum while you/your dad goes out, that it is not enough and it is simply not possible to meet their needs in a domestic home. I do not know if this is the case with your mum, but you do have to be realistic and be open to this possibility
If she does need a care home, then you can start looking around for somewhere else. Care homes are not all alike - they all have different criteria about what they will, and will not, accept, so be brutally honest when telling them what your mum is like. You do not want to move your mum somewhere and within a few weeks have them turn around and say that they cannot meet her needs, so you will have to find somewhere else. Ask about whether they would accept the LA funding - dont get yourself in a position where you will have to pay "top-up" fees. Look beyond the decor to see the care provided. My mums care home was very shabby and old-fashioned, but the care was excellent.