Welcome to the forum.
When your mother's money runs out - actually, when her assets drop below £23,000 - the LA will start to contribute towards her care home fees. Some homes won't accept the rates that the LA pay and so do bear that in mind if your mother's assets won't last very long and she will need LA funding. Some homes will accept local authority rates if the person has been self-funding for a couple of years before they require LA funding.
In terms of choosing a home you could: (1) ask around locally, (2) speak to the elderly care nurse at your mother's surgery who might have knowledge of local homes, (3) ask Social Services for recommendations, (4) look at CQC inspection reports and (5) look at the carehome.co.uk website which is a bit like TripAdvisor. Do bear in mind that homes can change quickly and so views can become out of date.
Do visit several homes as homes can vary a lot in terms of facilities and atmosphere. Some are like hotels, some have a clinical feel and some are homely. It's easy to be attracted by facilities but do consider whether your mother is likely to use them. The quality of the care is more important than the quality of the decor and furnishings.
Visiting a care home full of people with dementia can be a bit of a shock. If you visit in the late afternoon you are likely to see the home at its 'worst' as residents will be sun downing then. Activities are usually offered in the morning, and in the early and mid afternoon you may find most of residents asleep.
It feels a big responsibility choosing a home and it's natural to worry about whether you are making the right decision. If your mother is self-funding you can change homes if you're really not happy with the one she's in.
Do keep posting for advice and support.