Well, I suppose you've helped out a couple of times...but I'm embarrassed at the idea that your poor mum actually asked for help and was then refused. She must have been desperate before she got to the stage of asking.
I was my late husband's carer. There was only an issue of changing bedding and so on if he became ill on top of his chronic condition and I did everything myself. (At one point I was working full time and also caring for my mother - but I did have carers coming in to help with her.)
The one time that I turned to family for help was when I was waiting for test results - I'd had bad news following two tests and had to wait three months for the next one, so started to put things in order, just in case.
I phoned DH's daughter to let her know that if anything happened to me that she and her brother would have to organise care for their dad. Her response? "You're not thinking of leaving him are you?"
3 months later, I got the all clear from the doctor. I phoned the daughter to let her know, but reiterated that if anything happened to me, her dad would need care. I didn't expect the kids to do the hands on stuff themselves. I knew that there was no chance of that. I just wanted someone to advocate for DH.
Her response this time was "But you're all right, aren't you?" I never forgot and I never asked anything of her again.
Your mum won't forget, OP.