I have a relative that lives in a small apartment but it's quite messy. She pointed out that her home was messy and explained that she couldn't clean it because her health wasn't up to the tee. How could I appropriately approach the subject of offering to clean it for her without causing any hurt feelings?
I had a relative who was injured and wife had cancer. When we visited, we talked about their situation and the things that they both wished they could do.
Some people take inordinate pride in their housekeeping and see inability to keep up with cleaning as a sign of failure. So I wouldn't approach it from that angle.
With the relatives above, I offered to cook a meal for them. That was something that they knew they needed and neither had the ability to do. then, after cooking, I was able to say, "Hey, I made a mess of the kitchen. Let me clean my mess up." No one will say, "Hey, please leave me a mess.." So that worked out well. Then I could give the kitchen a good cleaning, and she was happy that I could do that while not insulting her housekeeping.
After that, the discussion got harder. I offered to pay for a maid service, but was told that he would not accept that. So there's an element of pride associated (again) with that. However, once successful with the kitchen, had I wanted to, I would have said something like, "Kitchen's done - can I give you a hand in the living room? Is there anything you need moved around? I've swept the kitchen - how about I vacuum the living room?" I would keep the offer small- don't insinuate the whole place needs cleaning, but ask to do small tasks and keep going.