When I first launched the game, it asked me if I wanted to download the DLC (after the update). Then it turned out that I didn't have enough space, so it quit the install, after the download. So I deleted some stuff and tried again. This time, it didn't tell me anything was wrong until after trying to install, now I got an error message. Went to the data management anf it told me I had "unnecessary files" that could be deleted. The screen got stuck on "deleting, don't turn off the console" for a few hours, then I turned the console off. Went into the data management again, ignored the message about unnecessary files and deleted the MK8 update, along with some more files to make sure the DLC would fit. Started MK8 again, this time it didn't ask me if I wanted to download the DLC. But when I chose one of the new characters, which were no longer possible to use, it took me to the download screen, I got to choose whether to download pack 1, pack 2 or both. Finally, everything worked!
Long story short: Try to delete your downloaded extra content via the data management, then start MK8 and try to choose one of the new characters. It might work!