OK Wii wont turn on.
Like was said if it is under warranty, best bet is to send it in.
You can do some small troubleshooting (of course if you feel confident in doing this).
I would first test the power supply, simply get a multimeter and test the PSU input (to the Wii) plug. I am not sure what the Wii uses, but I'll assume its output will be DC 12v and the wii internally steps that down (through design) to whatever the components need.
If its the power cord, then that's solved.
Now if your not under warranty, its time to crack that Wii open. You want to avoid a ESD. So ground yourself out or risk your Wii.
Fact about ESD. You feel a static shock at roughly 6k volts. You see the static arc at around 20k volts. To fry a board it only takes as low as 0.5 volts! If that does not make sense do some reading about Volts and Amps and ESD. Amps BTW is what will kill you if your not careful.
ESD
Volts
Amps
Now...
Off the bat check for "bloated" capacitors. Most electronics use extremely cheap components, and it wouldn't be the 1st time I have seen the CAPS go. So if you see the tops look like its bloated, remove them and re-solder in new ones with the
exact same specs.
Without going into detail capacitors are used for not only retaining a charge, but to ensure proper levels of voltage and amps is steady through the circuits.
Then check for transistors that are burnt out, but in my experience CAPS go first and the most. Same deal, discharge yourself and replace with the same spec trans.
If its chips and stuff, depending on if you can get a replacement, you can redo those, but troubleshooting chips is difficult at best.
At that point (depending on your skill level) its time to contact a professional.
MD