Sorry OP. Generally I am very sympathetic to women carrying most of the load and it gives me the rage. However, there's something about your post that makes me feel sorry for your DH.
First the facts. He works full time and is an active father, a pleasant, caring guy, and sounds like he does atleast some stuff on top, if passively. So not totally uncontributing or useless. And he's been through cancer. That's a huge, traumatic deal. And if it's just a year later, and he's lost his fertility... probably both physical and psychological damage he's still dealing with.
Then there is your tone. Honestly, it doesn't sound like you like or respect him or care what he wants at all. You are basically keeping him around because you are obsessively committed to getting another child for yourself. And you are going to a sperm donor for it. I wonder how much voice he has had in any of this and how he truly feels? His job is basically to provide shelter and income for you to pursue your dream for more children, which won't even be his biologically? I may be totally wrong, but if he already feels dragged along on this ivf/ sperm donor journey, which is for you not him, while trying to recover from cancer, while working full time and being a good, loving dad. Well, frankly, I can understand why he might lack the energy or will to also do housework the way you want.
You don't sound like you have much interest or caring about his needs or emotional situation at all? You certainly haven't mentioned his desire for a further child or how he is, etc.... ?