She has been intervening when I would try to set up a cleaning schedule, and then nothing gets done like it should. I don't know what to do—I don't want my children to think that this is how people should live.
I'd suggest two steps here:
Have another chat with your mother-in-law. She is obviously doing a great job babysitting the kids, but you need to be really firm with her about modeling the right behavior for the kids. So, instead of making this about the office, make it about the kids' toys.
Suggest you get a couple of bins for the toys and that, at the end of each day, your mom work with the kids to put the toys away where they belong. Make this a normal part of their playtime.
Suggest to your her that she could use the same principle with the home office as you want used with the kids. Help her find a good filing system
(maybe a file cabinet) and some trays for her desktop to manage the mail, bills, etc. Then, each day as the kids are picking up their toys, she can
use those 10 minutes to do a quick pickup and tidy of her desk.
You win, the kids get a good lesson, and your mother-in-law learns that it's all about small, consistent steps.