If some reason your Wii U can’t connect to your WiFi access point, the root cause might be IPv6…
This is what happened with our 3 years old TimeCapsule unit. Our ISP is now offering IPv6 and I configured it on the WiFi router. It works like a charm for the Mac laptops and suddenly Wii U was no longer able to connect to the WiFi SID.
Turn IPv6 off and Wii U connects again.
Tested with Wii U firmware 4.0.0E and TimeCapsule 7.6.4
PS: please note some posts advice to enter device IP, gateway IP and DNS manually. It’s offer better to use your ISP DNS and not public one to take benefit of the download speed on a very close server.