because you are a woman and therefore better at everything ?
I know how frustrating it is, and I wish I knew the answer.
My DH doesn't sound as bad as yours as he deals with workmen (mind you, he knows plumbers, builders etc anyway through his job), all his own car stuff and odd bits of housework, though it can take prompting. I do virtually all cleaning, cooking, shopping, finances, holiday arrangements etc etc. We both work full time (no DC yet) and make an equal financial contribution. I leave the house 1-2 hours before him & get in about 2 hours earlier - the time I am out and he's here, he's asleep; the time I get in and he's still out, I'm cleaning, washing, cooking etc. He does work Saturdays though, which is why I ended up being responsible for so much, but I am beginning to feel like he should do more stuff for 'us'.
We will have to have a re-think if when we have DC and there is more work to do.
What I have found works is to give him jobs that are his responsibility, eg cleaning up dinner stuff, rubbish, car washing, and always thank him for doing them. OK they are sometimes not done as often or as well as I would like, but at least they are done!
I can't really blame him if he doesn't leap up and offer to clean the bathroom or whatever when he has got used to me doing it.
It'll be hard to change his habits, especially if things have been like this for a while, so i think you should start small, one or two jobs at a time.
Good luck, and if you find something that works, let me know!