It may be worth it to open it and see if the wire is disconnected. You may be able to reconnect it (may have to solder it). This is almost certainly the issue unless it was dropped and landed on the button somehow....I have a hard time imagining how the power button could be damaged.
If I remeber right it is pretty easy to mess this up when you take the wii apart if you are not careful, previous owner may have messed it up.
A gamecube controller is pretty cheap and easy to find if you want.
If you take it apart and are unable to repair it you can get a replacement part that should fix the problem
These guides should also be helpful
The light is red-off, green-on, yellow when it's off but wiiconnect 24 is enabled. If its off a single press should turn it on. I think to turn it off you need to press and hold. Check if it works (tactile feedback be damned) and if it doesn't check the wires and a last resort replace the part.