Nintendo tech forum user GAMEDEV posted a guide to manually connecting your Wii U to wi-fi. We've reproduced it after the break. We hope you're able to get online and start "Yeah!"-ing your friends' Miiverse posts immediately.
1. Go into connection types, Hit "Manual Configuration"
2. Input your SSID
3. Input the correct security type, this is key.
4. Put your password in properly
5. For ip address "Do not auto obtain".
6. open up cmd in your computer, type in ipconfig
7. Now input that ip address but add 10 to the last number so "192.168.0.1" will be entered as "192.168.0.11"
8. Enter the rest of your info inside such as subnet mask and default gateway exactly as you see it in your ipconfig.
9. Now hit next, hit configure now for your DNS. When asked about DNS hit "Dont auto obtain".
10. For the first server put in 184.108.40.206
11. For the second server put in 220.127.116.11
12. Hit Next, when asked about a proxy hit no proxy
13. Leave mtu as the default
14. it should work, if not check the ip and that it is the internet ip + 10. Good luck