It sounds like it's all been very stressful.
Number one most important thing for you to do is chase your local authority. The GP has no power in these situations and it's as good as a wasted phone call chasing them to chase social services. At best the GP will send an email requesting an assessment and this will be added to your mum's electronic file.
Ring the local authority, state she needs a Care Act assessment and possibly residential care. Ask for an estimate of how long they think it will be.
Do not release any equity in your mum's property until you have had both the care act assessment and the local authority financial assessment. Equity release schemes generally are usually a bad idea in these situations.
The local authority will 'back date' any funding your mum is eligible for from them to the date you first contacted them. If she is still in her own home they only consider her savings and not the value of the property. They are highly unlikely to back date 24 hour live in care. At best they will back date the equivalent cost of their rate of residential care (as it doesn't sound like she has any nursing care needs).
As part of the care act assessment the social worker will do the checklist for CHC. If the checklist is positive then they will contact all of the relevant people at the NHS to arrange the DST. You don't have to do anything.
Cancel the solicitor. There is no point engaging the services of a solicitor until you know if there's even anything to fight about. It sounds to me like your mum is unlikely to meet the CHC criteria if she's manageable at home with a live in carer and no nursing input, but the social worker doing the assessment will be able to advise you more.
Finally, request a carers assessment for yourself (at the same time you chase the care act assessment).