I have found a solution to this issue that doesn’t seem to be listed on the official support page. I sent them a form thingy to add it, but regardless I think I will post it here too in order to get it spidered by Google.
First, check out that page and make sure you’ve tried the solutions listed, especially unplugging the power supply for 90 seconds. You probably don’t need to do it that long, but unplug both ends and let it sit anyway.
If it still won’t power on when you plug everything back in, unplug the power cable from the Wii, and then unplug your video cable. Plug the power cable back in and power on. If it works, plug the video cable back in.
This probably hit me due to having a weird setup with a third-party Wii->DVI cable (which surprisingly handles 720p, but it’s also a PS3->DVI cable too) and a DVI KVM switch, plus my dad was fiddling with live electrical wiring on the same circuit-thing today. So yeah.
Anyway that might help someone else, maybe.
Addendum: I just hit this problem again, with the official HDMI cable this time. Not sure how it resolved this time, but I would check ventilation to ensure the unit is not sitting on a carpet and to let it sit, plugged in, overnight.
Addendum Addendum: Problem ended up getting worse over the months, eventually the Wii U was dead and I had to pay $200 to get it repaired (one of the boards was bad). Moral of the story: Send it in while your warranty is still good. Also use the copy feature to backup your saves before it dies.