You probably won't like this answer, but it's the only one I've found in 20 years of marriage: I just gave up on him ever helping around the house.
Seriously.
When we were first married, this was the only thing we ever fought about. In two full decades of marriage, it's the only thing we've fought about on a consistant basis. Over the years I tried EVERYTHING: nagging, scheduling, doing it together, begging, logical arguments, rewarding help, letting it pile up, tears . . . everything worked for a week, nothing worked longer than that . . . and worst of all, it caused massive resentment on both sides. He WILL DO whatever I fuss about, BUT it always leads to a fight, and he NEVER does the same task again unless I continue nagging him. Having a load of dishes done is simply not worth two days of resentment on both our parts.
The simple fact is I have a problem with the house looking like a pigstye -- he doesn't. He will continue to leave trash on the counter right next to the trash can because it just doesn't occur to him that it matters. He will never wash a pot that he doens't want to use RIGHT NOW because he sees no point in putting forth the effort. He has literally never vaccumed the floor, and has never cleaned the bathroom. He considers clean clothes nice, but not necessary. I can't change this about him.
He doesn't leave these things because he figures I'll come along and "fix them"; it's not laziness. Rather, he just doesn't care whether they're done. He doesn't notice (or care) when I do them. He doesn't notice (or care) when I don't do them.
VERY IMPORTANT, THE KEY TO WHY I PUT UP WITH THIS: Except for this problem, he is a wonderful husband, a loving father, a good provider. He has a great sense of humor, lets me have my way on many things, always knows what to get me for my birthday, takes me to great places, sets a good example for our children (in all ways except cleanliness), shares my values, never questions what I spend . . . the list of positives go on and on. But he doesn't give a **** about whether the house is clean.
If this doesn't-give-a-****-about cleaning attitude was one of many problems we had, that'd be one thing. But honestly it's the only thing I'd like to change about him.
So a couple years ago it finally hit me that I have two options:
1. Continue to nag, feel resentful, and fight about the household chores.
2. Just do them myself.
I chose to just do them myself. Yes, sometimes I am resentful -- and I let him know that -- like when things really pile up -- but usually I remind myself of all the postive things he does, and I accept that "giving him a pass" on this cleaning is the best choice for our marriage. Doing it myself takes significantly less energy than fighting with him.
When it comes to things I can't do (i.e., fixing the broken ceiling fan or the leaky faucet), I give him a week and then I call someone. If he doesn't get it done, he's not allowed to complain about the money. He accepts this compromise.
No, it's not right. Yes, it's disrespectful. BUT I'd rather have him and this one problem . . . than to be without him in a spotless house.