I hosted the 5.5.X kernel files, so everyone can launch the kexploit / loadiine without anything else besides a Wii U

1) Make sure you have every file you need on your SD card. You don't need any for the kernel exploit in itself, but you need some for stuff like Loadiine and Cafiine.(for most apps you need to put them like this: /wiiu/apps/[appname]/[appname.elf])
2) Put your SD card into your Wii U, turn it on. Launch the Internet Browser.
3) Delete the cookies.
4) Go to "http://thekawaiiserver.net16.net". If the website is down, try to access a mirror site (list at the bottom of this post). Close the ad if one appears.
5) Make a bookmark of the website (so you don't need to type the url every time ). Press the HOME button, and launch the Internet Browser again. (This step is optional, and you only need to do it the first time you run the exploit).
6) Click on "Kernel10" or "Kernel31" depending on your needs (use Kernel10 for Loadiine and the Homebrew Launcher). It may freeze; in this case, force shutdown your Wii U, and retry until it works.
7) If it worked, profit! You may now choose the app you want to launch from the list, and click on its name.

You need to *always* run the Kernel Exploit before you try to open an app (Loadiine, etc.). That includes between reboots, after you already launched another app, a game...

EDIT: The 5.3.2 files are available now! If you're using a 5.3.2 Wii U, select Kernel 10 (5.3.2) or Kernel 31 (5.3.2). If you have a 5.5.x console, select Kernel 10 (5.5.0 / 5.5.1) or Kernel 31 (5.5.0 / 5.5.1). If you have a 5.4.0 Wii U, select Kernel 10 (5.4.0) or Kernel 31 (5.4.0).

EDIT 2: Apparently, some people have trouble making the exploit / apps work. I think this happens because the Wii U can't load the files fast enough. I'll try to make a US mirror to see if it makes things better, but meanwhile, you can also use http://loadiine.ovh and follow the instructions on the page

List of Mirrors:

http://thekawaiiserver.net16.net
http://thekawaiiserver.96.lt
http://thekawaiiserver.xp3.biz
http://thekawaiiserver.byethost13.com/


Credits:

This guide: http://rhcafe.us.to/?page=t&id=108
MarioNumber1 & NWPlayer123 for the 5.5 Kernel Exploit
The guy who leaked the Kernel Exploit
@Kafluke for the 5.4.0 files