Hi, thank you for contacting JustAnswer.com. My name is Russell. I will do my best to provide the right answer to your question.
My first impression was, this is probably a case of 'broken Wii in need of repair', I'm sorry to have to say.
But you can try this first, before assuming it is indeed 'broken' and in need of either repair or replacement:
1. try it in a different room if easily possible... if it works there, it's an interference problem, likely. Which can be solved, if so... let me know if so.
And try this after the above: remove the battery cover from the Gamepad
Then turn on the Wii U via the Gamepad. And then push the sync button on the Wii U til u get the card suit code.
Then unplug the battery from the Gamepad (while it is still on), and wait 60 seconds.
Then plug the battery back in and turn the Gamepad back on, while pushing the sync button on the Gamepad (this should allow u to re-enter the code you see on ur TV screen).
And with that, the Gamepad & te Wii U should be now synced.
If the Wii is fairly new, it may be within warranty, in which case servicing would be free of charge.