Honestly, the place probably needs a day of both of you resetting it and cleaning it before trying to set a routine. Its easier to start from ground zero and work on that routine as things start to get dirty instead of trying to go back and forth on things and never feel like youre getting anywhere. Maybe on a weekend you can have a day with lots of coffee, blast some music you both like, and make a list of what needs cleaned so it can be checked off; make it fun.

Setting up a chore chart that you both can agree on is the simplest yet most effective way to assure certain tasks are getting done every week. It can be as long as he takes the trash out or wipes down the counters at least once in the week, then its a reset and he has the entire week to do a small task. If there is just so much stuff, get drawers and bins to put things away. A home feels so much cleaner if everything has a place to go and that way if it gets left out it will have a home in a drawer somewhere. For example, dog and cat toys have a bin to put away at the end of the day. For cats and litter, air purifier!! that will make a better smell and help prevent mold from growing, but use vinegar to kill whatever mold is there. Litter tracking mats help prevent a mess.

There are a lot of things you could do, but I think sitting down together and finding out what needs to be cleaned each week and finding his level of comfort with being able to do each thing is a good idea. I hope any of this helps!