Hi @colorwave ,
Here are two links that may be of some help.
This [https://gbatemp.net/threads/gameboy-color-blown-fuse.366969/|one] describes what to do if the problem is with dry solder joints on the switch. Scroll down to the section dealing with the power switch (it starts about halfway down the page) and check if it applies in your situation. (It is the link which is embedded in the link below).
This [http://www.instructables.com/id/Nintendo-Game-Boy-Color-Power-Switch-Replacement/|one] describes how the problem can be solved if it is not a dry joint problem and you need to replace the Power switch.
Between the two you should hopefully be able to solve your power switch problem
Here is a link to the ifixit [guide|12743] guide, which should assist you in gaining access to the motherboard so that you can check out which solutions you need to use from the links above.
Hopefully this is of some help