Why not go back to your mother's GP and tell him/her that you are doing all cooking, cleaning and shopping for her as she is unable to do this for herself. Perhaps you Mum could get outside carers to go into her say 2 or three times a day to do this and I am sure it will cost a lot more then what you are doing. GP are not necessary good with elderly people needs, as they have not reached that age themselves.
In my case my close relative was suffering from dementia this started in her late 50’s it was only after she lost her husband and then after a serious car accident and been in hospital for six months that it was dementia was diagnosed. Also she suffered from osteoporoses which meant that she was in dreadful pain continuously but it was not diagnosed in the hospital by the consultant. This condition runs in the family. She was transferred to a care home from the hospital and was there for 2 years year. She had many falls in the care home ending up with broken bones and was taken to hospital for x-rays but still not diagnosed with osteoporosis. It was only when she was in dreadful pain and a kind carer told me that she had these many falls with broken bones that I mentioned osteoporoses to them. It was only then that she got treatment for it. So there are many age related illness that are not picked up by the medial profession and I hate to say it but these people do not know how to assess elderly people as they have passed their 'sell by date' so to speak they cost government to keep and take uses lots extra resources, which now are unavailable to spend. Think of all the money that the government i.e., state pensions care allowance and all the extra benefits that would be saved IF we elderly kick the bucked soon rather than later, off my soap box now.
Also you could send a letter with the application to the welfare people saying that you are the only person available to do these things for your mother.
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