I worked (via Age UK) as a Home Help for nearly 10 years (absolutely loved it) and would do everything you describe bar cooking a main meal. As far as I remember, Age UK say Home Helpers can prepare 'light meals'. This would include heating up ready-prepared meals. Maybe start by having a chat with Age UK. Everyone is DBS checked and they would usually be able to supply cover visits in case of holiday/sickness. I was required to have Business cover on my car so I was insured to take people out (eg to shops, appointments). I would also second Home Instead as mentioned above, and I think they do more on the cooking side, but they are more expensive than Age UK.
Lots of clients wanted to carry on doing as much as they could while they still could. Sometimes we 'shared' tasks to make this work, eg they would strip the bed and maybe put it in the washing machine, I would make the bed (the ability to wrestle with duvet covers is one of the first things to go!), turn the washing on if they had forgotten how to use the washing machine (common) and deal with getting heavier damp washing out and hanging it up.
One of my last clients was in her 90s but absolutely loved ironing. When possible I would set her up with the ironing board in a low position in front of her chair with some small easy-to-iron items eg tea towels, napkins. This simple thing made her very happy!
I wouldn't describe the role as a Carer as I always assume this includes personal care, which I didn't do as a Home Help. I think Home Instead staff can also cover personal care. Another thing to ask about is whether they will deal with medications if required.
Good luck, and be fussy (within limits) - even with getting help via an agency, some people work harder than others, and some people will 'click' with your mum, or not. You might not find the right person first time.