This same problem has happened with three Wii Remote Plus/Nunchuck (WRP/N) combinations, two third-party (of different brands) and one Nintendo-made. There are two problems that occur: 1. the WRP/N becomes unresponsive to all user input and instead sends MK8 commands of its own, resulting in behavior as if the nunchuck's control stick is jammed into one of the corners of its housing, and 2. the WRP/N disconnects completely, usually after (1) and the Wii U asks the user to "Press A on the controller you want to use.
Before you ask, the batteries have been replaced many times over in the throes of testing this, and the Wii U has been powered off and on many times over, along with the controllers, which have also been re-synched many times over.
I have searched the internet for similar problems have not discovered anything useful. However, I did find out that Mario Kart Wii may have had the same issue: http://www.mariokartwii.com/threads/108986-Wiimote-Nunchuck-problem.
Does anyone have any ideas as to either what's causing this or how to fix it?