I agree with Martin. It is not possible, long term, to run two households without help. You need to delegate as much as possible to paid carers. Carers could help with washing and dressing, give your relative breakfast and perhaps a light lunch, wash the dishes, administer any medication and wash and dry your relative's clothes. You might also think about engaging a cleaner. Your relative might be resistant to outside help but you need to persist, otherwise you will get burnt out and won't be able to do anything for him/her. Believe me, even with carers in you will still have plenty to do as there is much that cannot be delegated, particularly in relation to your mother's finances and the upkeep of the house.
Another thing is that if you spend all your time with your relative rushing around doing chores then you will have no time to spend just being with your relative, which is what s/he wants.
Your relative should be able to claim Attendance Allowance (which is a non-means tested benefit) and might be able to claim a Council Tax discount or exemption. This will help you pay for care if your relative is not entitled to have the full costs of the care paid for by the Local Authority.