Here's a thing to think about: Levels of cleanliness can be improved. People can get better about picking up after themselves and doing household chores. But you can't do that if you're stuck in the middle of a major depressive episode and can barely get yourself out of bed in the morning.

Now, full disclosure I was never a "filthy slob", but I used to hate doing chores and would put them off as long as possible. Some stuff I just wouldn't do. I was also extremely depressed. It's taken time and a lot of effort, but I've gotten to the point where I keep pretty much the entire house clean and in order without a problem. I still struggle when I'm having an episode of serious depression (I have chronic MDD so it happens) because depression interferes with your ability to motivate yourself, but I've developed much better habits.

It's possible that OP could help his partner get better at cleaning up and keeping things neat, but any efforts are unlikely to work until she's had professional help for what looks like some very serious post-partum depression.