who carries out the mental capacity assessment? So far we have only witnessed her being asked acfew questions about where she is and what day it is and who is the PM
what you will have witnessed is a memory test (often referred to as a mini mental) which gives medical practitioners an idea of what someone's memory is like at a particular point in time. It can be a useful measure if carried out at different intervals (e.g every six months) to measure a memory decline.
mental capacity is a little different and is decision specific, it asks can the person understand the decision that needs to be made, can they retain the information they need to understand about the decision long enough to weigh up their options and make an informed decision.
anyone can carry out a mental capacity assessment, but for decisions about going into a care home it's usually a social worker who does it as part of a best interests assessment, but also sometimes a nurse.
if your mum is able to demonstrate that she can understand the risks she may be at staying at home, and can explain why she would prefer to take this risk rather than be 'safer' in a care home then she would probably be considered to have capacity.
if it's felt that she lacks capacity then the organisation responsible for funding the care can make an offer of what package they feel would meet the assessed needs. If this is 24 hour care then it's likely that the package offered would be residential care rather than live in care at home.
if your mum or anyone else wanted to privately fund the extra amount between a care home and live in care then this would be an option.
again it all comes down to who has what in the way of LPOA though.