Rat caught (dead), but it was living in walls - how to clean up?

TLDR: How to clean up after a rat has been in the walls.

100+ yr old home, Illinois, urban area. New to the home as of last summer.

I've had a rat in my walls for more than a week (I know). It was in the walls, in the space between floors, and in the basement. I had live traps out in the basement (which it outsmarted).

This morning, I found the rat in my basement, lying dead on the floor. I have no idea how it died (I hadn't set out poison, but maybe the previous owner did?) That's its own set of concerns.

So what is next? This rat was in my wells pooping non-stop. Is that just how it goes?