First: Your power brick is most likely NOT broken, don’t waste your time opening it up, instead unplug it, go outside and run around for 15 minutes, come back and plug it in and see if it’s now magically fixed! (Shortcut is to unplug it and short out the output to drain the capacitors in the power supply, making damn sure it’s unplugged before doing so so you don’t actually pop the fuses)
On with the opening up step if you’re still not convinced. . .
If you’re wondering how to open a Nintendo Wii power supply brick you’re in luck. For some reason my Wii wouldn’t power on at all last night, no red, yellow or green lights on the console, nothing. I grabbed my trusty Harbor Freight multimeter and checked the power supply output. Nothing. Next up was to open up the power supply and find out if my power brick had suffered from a blown fuse. Of course, Nintendo put proprietary screws on the power brick to prevent someone from opening it, and of course they weren’t tri-wing, they’re some nutball spanner screw heads. The good thing is that they’re a terrible “tamper proof” head, as they put the dimples directly opposite of each other (even number of engagement points).
If you want to build a truly pain in the ass fastener you need to have features that don’t line up, typically by using an odd number of engagement points, like the tri-wing screws they use on the console. Just grab a pair (or two) of needle nose pliers
And span the screw with the tip of the pliers, apply force to engage the pliers in the dimples and then carefully twist left. You should be able to see the screws almost immediately rise away from the supply. The good thing is they used correct, coarse threaded high angle plastic screws so it’s only about 4 turns to get them out of the supply.
The annoying thing is that the fuse wasn’t burned out on the supply, nor was there anything else wrong with it. It definitely smells like hot plastic/capacitor electrolyte in there, and they’re using lead free solder and don’t have the best joints I’ve seen, but nothing major. I’m thinking the supply may have not started up correctly and therefore shut it self off (there are some optocouplers on there to detect if the supply is operating correctly and feed back to the mains side of the supply). So most likely by me unplugging it and the time it took to disassemble “fixed” it, otherwise there may be a dry solder joint in there that was flexed the right way to remake contact while I was taking it apart. YMMV
Nintendo has a remarkably useful write up here regarding resetting the Wii power supply.