Wii U not recognizing the adapter + ethernet cable

This Microsoft support page lists three reasons why you might get a red light:

• There is too much current.
• The voltage is too high.
• The temperature of the power supply is too high.

You say it happens straight away so that rules out #3.

Have you tried it in a different socket?

If that still fails then you might have to go for Solution 4: Order a replacement.

If the power supply works in a different socket then you need to get your electrics checked. There might be a problem with that socket.

I have the problem myself and only have seen it in Skyward Sword, none of my other games show graphical issues. Some digging online shows it to be a hardware issue--your Wii's GPU is a bit fried. Not enough to ruin the graphics in other games, but since Zelda runs a more advanced graphical engine and pushes the Wii harder with depth of field effects it causes artifacts if you have a bad GPU. Nintendo can repair it, and that should solve the problem, but it may be a bit costly.

One thing to consider is the always-on nature of Wii's internet connectivity. Having the Wii on a constant 24 hours/7 days a week can bring it to a surprisingly hot temperature and probably wears out the hardware faster. Perhaps this touted feature was a bad idea, I can't help but wonder if doing this for 6 years caused my Wii's GPU problems.