What's the easiest way to clean the brake deposits out of the corners and on the back? I believe these are painted wheels.

I'm thinking of trying a "magic" spray on and rinse off product such as these:
Simple Green® All-Wheel Cleaner
P21S Gel Wheel Cleaner

If that doesn't work then maybe one of those round foam balls you use with an electric drill.

The easiest way is to use a proper acid based wheel cleaner. There's a product called Bilberry that is phenomenal but it's sometimes hard to get in the US. Chemical Guys also has a few different gel type wheel cleaners that work very well, though require more elbow grease than acid. If you have a detail shop nearby, they will likely sell you a bottle of the acid if you ask. It's not going to be cheap but it will do all the work for you in a matter of seconds.

Do you know what kind of acid it is? What concentration? "It works in seconds..." then you brush &rinse away the acid & dirt? What if paint is partly gone, exposing the alloy to the atmosphere, and to the acid cleaner?

From the Bilberry wheel cleaner website:
This product is NOT suitable for un-laqueured polished rims

Easy-off oven cleaner works great. It will take the coating off as well if it is still on the wheels though

Sodium or potassium hydroxide (lye) and aluminum / magnesium alloys don't mix too well, though.

I have used an orange spray cleaner .that works very well.& also Dr. Bronner castile peppermint soap..both are very gentle & non chemical crud containing..