I'll echo what PPs have said - you have to find what works for you. And hopefully you can get ideas from others to help figure it out with less pain, so I'll share what works for us. "Not doing the housework and waiting to see what happens" doesn't work for us. DH has a tolerance level for dirt and mess beyond my wildest dreams (and I'm nowhere close to tidy!) and doesn't think you need to do laundry until you take off your last pair of dirty underwear. So, we live by lists. Every morning I leave him a list of what has to be done today and what needs to be done soon. I get that you can't always do what you plan when you have a baby who needs you, but ours still naps every day, so I know he gets at least an hour, and the list is never an hours worth of work.
In the end, we split a lot of stuff, but I view that DH definately has primary responsibility for cooking and cleaning during the week (and that doesn't mean everything gets to wait til the weekend!) I may do a few things before I leave for work or in the evening if DD is in a good mood and wants to help, but I really view that as my time with her, so I don't ever put the housework before her (except that I do use housework as a stress outlet, so sometimes after a really bad day I do it just because I need an outlet). And then, as I go to bed, I tell DH anything that I view as a "must" get done that night (he stays up after DD and I go to bed).
The key learning for me was that I have to tell DH what I want him to do (only with this stuff - the parenting is completely natural to him and he, in fact, often tells me what to do
). I refused to do this for a long time because he is a super intelligent man with able eyesight, and I was determined that he should be able to see what needed to be done and do it. And then one day, I thought "what the h*ll am I fighting this fight for? Tell him what you want and see what happens" And so here we are. I've also learned that there are some things he just is not good at (e.g., making a grocery list and doing meal planning). So I make the list, he goes to the store, and then I leave him a list of meals for the week and he picks what to make each night.
Good luck! Our first few months were rough with regard to this, and we still occasionally have it out (in a very healthy way, of course!), but we learn a little more each day and it does get better!