In her hard-earned leisure time, my wife occasionally enjoys playing on our Wii system. She came to me the other day and said it wouldn't turn on.
I had a look and indeed the Wii appeared to be deceased, so I removed it to my office. First I plugged in the power supply and checked the leads with my multimeter. No voltage. Then I attached a 12 volt battery to the Wii, and on it blinked and the optical drive jerked to life. Problem isolated. The next thing I did was - what else? - go to Google to find a solution.
I'm very glad to have found this blog post by Damon Kohler: Is your Wii power supply broken?
Mr. Kohler had the misfortune of taking apart his entire Wii power supply, only to find there was nothing wrong with it. He put it back together and it worked. He suggested letting the power supply some time to rest; that is, to lose any residual charge in its capacitors and subsequently any memory it might have. He also recommended shorting the pins to help guarantee a reset.
I did this, and seconds later the Wii was up and running. My guess is the power supply has some kind of surge or error protection that causes a shutdown. Or maybe it's just a design flaw. Anyway, before you take apart your power supply or go and buy a new one, make sure you try the above.