What do I need to play Disney Infinity 3.0 on Wii U

We finally found the answer. You must have a Nunchuk connected to the controller. If you don't, it doesn't tell you in any way. It just won't connect at all.

Also, I found that it never works on the Wii version, except in Toy Box, but that limitation does not apply to the Wii U.

Yes, but note that both games require the Nunchuck attachment as well.

Requirements

• Twilight Princess requires any kind of Remote, and a Nunchuck attachment.
• Skyward Sword requires a Remote, MotionPlus, and a Nunchuck attachment.

Controllers

• There are two versions of the original Remote (or "Wiimote"):
  • The original Remote has more limited motion capability, but is sufficient for most games on the Wii.
  • Later versions, called "Wii Remote Plus", have the MotionPlus attachment (below) built-in.
  • Though some versions of the Remote might be branded "Wii U", all Remotes work with both the original Wii and Wii U, including when playing Wii games on the Wii U. There is no Remote that only works with the Wii U (though the Wii U does have its other stand-alone controllers exclusive to it).
• The Nunchuck is an attachment for the second hand, and is necessary for many "traditional" video games on the Wii.
• The MotionPlus adds additional sensing capability to the original Remote, and also has a pass-through for additional accessories (such as the Nunchuck).
  • Any game requiring a Wii Remote with the MotionPlus attachment can instead use a Wii Remote Plus.

As the controller you've linked is a Wii Remote Plus, that means it can satisfy both the Remote and MotionPlus requirements of Skyward Sword (and will function like a regular Remote for Twilight Princess). However, if you don't have a Nunchuck, you'll need to get one in order to play either game.