The council usually fund care if the person needing it meets the criteria.
They will only find a certain amount of hours, depending on their assessment.
If your mother needs 24 hour care, they will not fund this.
What you need to do is first speak to the care home to ask them to give you as much detail as possible to her care needs. Is she needing actual nursing care as in nurses or are carers adequate for her needs? If she needs actual nurses, there is certain things community nurses can doz this would be set up with the home/doctor making a referral to community nursing. They are able to carry out a great deal, but they are on time scales.
If carers is adequate the maximum the council usually will fund is 4 visits a day. These aren't long visits though.
You could fund a private carer, but it is costly. If you have the funds to do this, then the carer would need a detailed assessment.
Failing that, what my family did for a dying relative was we had a carer package that visited 3x daily, a community nurse visited once daily, and the rest of us family had a rota so that one of us stayed every night. Me and my mum were trained by a specialist nurse how to administer medication through a PIC line, I took on showering, nail care, skin care etc so that the carers didn't have to factor it in to their Time.
We were able to adapt the living room into a bedroom with a hospital bed, hoist, commode etc
So it is doable, but depends on the level of care your mother needs, and if these needs can be met at home, and if you have the support around you.
If I can help with any questions just ask.
For reference; I was a senior carer for 10 years in a home, gave cared for two relatives with end of life care so I do have a little understanding and knowledge.