I'm sorry to hear about the cancer; I hope your treatment is successful.
Given that your illness is so serious, I think that the question of whether your children are old enough to help is a bit of a side issue. They will probably be very worried about you, and that fear combined with teenage hormones and a stressed father , plus arguments about housework, might be more than they can cope with. I understand the point about everyone mucking in, but in this case I think dealing with everyone's emotions running high should come first.
If your children were to go to your MIL, she would be able to keep them busy and provide reassurance when needed, and you would have no worries about their safety or whereabouts.
If only your DH is home, and he's working during the day, very little mess would be generated. If everyone pitches in to do a big clean before you go then there will really be nothing to do with nobody there, apart from your DH clearing up after himself as he goes along.
It also means that your DH won't have to worry about getting food in, sorting out meals with your children etc if they are with MIL.
The time to make the point about helping is when you're home, I think.