Hi Suzy
So sorry to hear about your mum refusing care. It's very frustrating when you're trying your best to keep her at home. My mum refused to let carers help her for the 2 and a half years she was at home after dad died. We kept the support in place so we knew mum was at least letting someone through the door to check she hadn't fallen etc. Eventually mum moved into a care home as she was increasingly lonely and forgetting to eat. My sister and I have been amazed at how well she has settled and allowing carers to shower her. Perhaps this is because she doesn't see it as "her territory".
Some people have suggested saying something like the Dr has said the carers have to help, or maybe that the carers are training and have to do X number of hours on tasks before they can be signed off?
Does she think that she has already showered and doesn't need the help, or that she doesn't want them to help? Does your mum ever let family members help? My mum would willingly let my sister and I help, but not want carers in which would have eased our workload and enabled her to stay at home longer.