I can’t remember how many hours you said you were doing per week, but if you are only getting £2 per hour, then if you were to even just do those hours in a NMW job, you would be getting an extra £10.21 an hour. So if you are doing 40 hours per week for your mum that is approx £408 per week in additional wages (pre-tax) that you are missing out on. So for a month, that is £1632 (pre-tax) additional wages that you are missing out on. Or look at it another way, that you are ‘giving’ your Mum. Can you afford to ‘give’ (subsidise) your Mum £1632 a month??? If you had that extra £1632 per month, what would you do with it? I imagine a good proportion would be building up savings / pension pot for your future.
Getting a social care assessment may make your mum actually face what things she can actually do herself - as she will have to justify it to an independent person - the assessor. She will never probably admit these things if it is just to you (as she wants to have your attention / help, as she is lonely). They will essentially come up with a recommendation about what care your Mum needs. Maybe it will be a care visit once a day, to provide support with a hot meal at lunch time (and maybe making a sandwich tea that your Mum can have at tea time)? Maybe also making a hot drink in a sipper flask that your Mum can drink throughout the afternoon?
It may work out cheaper for your Mum to then also have a cleaner (they can come once or twice a week, whatever your Mum prefers), and a gardener, rather than relying on carers to do household tasks (if she would rather have the carers focus on meals for her).
There are also charities that can allot a certain number of hours for a person needing care, and someone can come over for a couple of hours to keep them company, or take them out.
It will actually be better for her as well, in the long run, to outsource the ‘daily grind’ parts of her care to a paid carer / agency, whether that is paid for or subsidised by the council, or she pays for it herself. Ultimately, it will avoid family members (especially you) having ‘care fatigue’ and withdrawing from her substantially / or having a poor relationship with her, and it would mean that instead of using your precious hours to do the housework / gardening / cooking etc then you could devote the 4 hours that you would have spend on the garden actually taking her out to a garden centre for lunch, of to the local stately home / gardens, or taking her to a local social group / drop-in sessions, or even just sitting with her chatting and having a cup of tea / doing a jigsaw or whatever.
If she can face up to keeping whatever independence she can get - and doing / planning things strategically to get more efficient at managing daily life, then her relationships with family will improve and she will be less lonely.
And your situation will massively improve. You need to consider what time you are willing to commit to her on a weekly or monthly basis as a daughter (not a carer), whether it is more regular ‘popping round’ for an hour to visit on a few days a week, or whether you have a particular afternoon that you go over, and plan to do something nice with her. As she has mobility issues (with a walking frame), it may mean that she needs to get a folding wheelchair for trips out with you. But there are plenty of taxis available that can accommodate a folding wheelchair in the boot.
Sorry for such a long post - but changing things now (and being firm that you are no longer able to subsidise her care costs by £1600 per month), will have a massive improvement on both of your lives. She might say that she is not willing or able to pay for a high level of care - but she is essentially expecting you to pay for it (in the opportunity cost of you having to work for £2 an hour, rather than £12.21 an hour).
My parent has 2 carers, 4 times a day, and the total cost of this is around £2400 a month, but she only pays a proportion of that (as it has all been arranged via the adult social care team). It was a big transition for my mum (and one that she was very unwilling to have, at the start), but in some ways she absolutely loves it now, as she has started to see the carers as friends, and she really gets on well with them, and she really enjoys the social aspect of it.
Also - Wiltshire Farm foods, they have various meals like roast dinners that come with veg (in the different compartments in the ready meal container), and they are great. They also have things that are more like standard ready meals (like pasta, eg macaroni cheese) - and it may be cheaper to get the standard ready meal type meal from somewhere else (eg Iceland), and just use Wiltshire Farm foods for the more complicated meals (that come with veg etc). My mum has them - you can order them over the phone, and the person that delivers them can put them straight in the freezer for you.