I used to work in Nintendo Customer Service
Although it sounds as if you may have tried this already when you swapped outlets, try unplugging the Wii–without reconnecting it–for about 1 minute. The Wii, like the GameCube, has a built-in surge protector that’s tripped during electrical fluctuations and can only be reset in this manner.
If that doesn’t help, I’d suggest trying another AC Adapter if you can, just to narrow down the problem, as replacing the adapter is obviously much cheaper than the system. How long have you had it for anyway? If it’s less than a year (or somewhere close, Nintendo’s pretty good about that), Nintendo will repair it for free.