I have a feeling that both sides will feel they have the higher work load regardless of what the reality is.
Generally I do the dishes and take out the trash, while my wife does the laundry. Other cleaning chores get divided up as needed. Dishes is something that needs to be done daily and is 30-60 minutes on average. Laundry is once a week and takes most of a full day on and off. I also do all the outdoor work like snow clearing and lawn mowing.
I'm not sure how old your kid(s) are, but with kids under two all bets are off. Nobody gets sleep and everyone's on edge. We would try and trade off who got up with the baby, but it's never fun.
Otherwise, it just depends on what kind of partnership you have. If you are doing 90% of the work and are fine with it, then keep it as is. If you are doing 55% and mad it's not 50/50, probably more of an uphill battle. If your partner is super lazy and doesn't want to do anything, then you need to be honest and talk about it.
Another thought I have is, if you are both dug in and can't align, it's absolutely worth trying some couple's therapy. I'm just a dude on a gear forum, but my ex and I could never get on the same page and everything sucked at home. We split and my wife and I have almost always been on the same page from day one. Not to say we never disagree, but usually we can figure it out if one person is struggling. Because we both really give a shit and want everything to work.
Anyways maybe a place to start, see if you can find some common ground. Can you have 15 minutes of peace and quiet when you get home first, then take over with the kids so she can get a break for a bit? Can you wait until after the kid goes to bed to do some cleaning, that way you can make some time as a couple or family to actually enjoy life for a hot minute? Switching off bed times is a great idea, as is just generally having a schedule. Having one night per week where you can get out of the house, then another where she can, and maybe 1-2 times per month trying to have a date night.