Is there a proper or safer way to clean kool-aid out of a wireless keyboard?
My plan is to take it apart and clean as much gunk off and out of the keyboard inards as I can.

The best way is to be careful :D You should be able to pop off the keycaps to get access.

Yes you can take it apart they are very simple. Just use care some keys fall out easy some don't. Disassemble carefully with keys on bottom just in case they fall out easily. You can rinse the whole thing under water just make sure it is dry before reinstalling.
I have taken several apart and cleaned them. Some had circuit traces ate away you can use conductive paint to replace trace, the same paint used to repair defrost grids in your car, it will work great.