Wii Power not working

Basically, I just got a new SDHC card and when I went to use the Wii, after several minutes I unplugged it from the wall to prevent an update. Then it doesn't work at all. Not even the red power indicator is on. AND it doesn't load anything!

What do you mean "unplugged it from the wall to prevent an update"? You unplugged it while updating? How did it start updating anyway? What version of the System Menu were you on previously?
From what it sounds like, you unplugged it at the wrong time.

This is why you should never unplug during an update. Sadly it is quite likely that your Wii is now bricked beyond repair. If it is quite an old Wii (more than a few years old), it is possible someone with an Infectus could fix it. Otherwise it is unlikely to be able to be fixed. Even if that is the case, you will have to find someone capable of doing the repair, ship your Wii to them and pay them for their services. Sorry I can't be of any more help than that.

the red power light should still come on even if you unplugged it during some important part of the update.

you may be able to reset the power brick. ive had to do this several times on both of my wiis.
[www.nintendo.com]

If the power light is completely off, not even red, it's a hardware problem. The system isn't getting any power. If you're lucky it's just the power supply box or cord that needs replacing.