You've got most of it covered here, but I'll put in a vote for Hyrule Warriors. If you like Zelda, it has enough Zelda lore to keep you interested, story-wise and plenty of fun-factor. I'd never played a Dynasty Warriors game before this, but I loved it. Metroid Prime Trilogy is also a must.

Other noteworthy titles: ZombiU, Wonderful 101, and DKC: Tropical Freeze.

Like you, I skipped the Wii generation but still found that the library was high quality if you sift through the shovelware. Wii U's library is hight quality too, just low quantity right now.

Edit: Bayonetta 2. I must be a fool to forget that one. 

Like so many others, I recommend Pikmin 3

Most already posted what I'd recommend but may as well reiterate:
- The Wind Waker HD
- Mario Kart 8
- Smash U
- Hyrule Warriors
- 3D World

Since you don't like Smash or Mario Kart, here's what I recommend:

Super Mario 3D World
Donkey Kong Country: Tropical Freeze
The Legend of Zelda: The Wind Waker HD
Pikmin 3
The Wonderful 101
Rayman Legends
Deus Ex: Human Revolution Director's Cut
Shovel Knight

As for the Wii's library, these are my favorites:

Metroid Prime Trilogy (you can get this for $20 on the Wii U eShop)
Super Mario Galaxy/Galaxy 2 (Galaxy 2 can be picked up on eShop)
Zelda: Twilight Princess/Skyward Sword
Xenoblade Chronicles
Donkey Kong Country Returns
Resident Evil 4 Wii Edition

Since you like some classic era Nintendo stuff (like me), the games that are must-haves are SM3DW and Xenoblade. 3DW is like classic 2D Mario given a modern 3D makeover. It's fabulous. I played Xenoblade on my Wii U and I was truly shocked at how good it is. It's like classic Final Fantasy with some modern touches. It's limited only by its SD visuals, but it's still beautiful. I expect the sequel to be amazing. I had never played Zelda WW on gamecube so playing it in HD on Wii U was a real treat.

You're wrong. It all depends on your tastes obviously.

I do not care for Jump&Runs, so no Mario for me. I like SSB and MK, but actually they didn't got a lot of playtime yet on my WiiU. I mostly played Monster Hunter and Hyrule Warriors (http://gamrconnect.vgchartz.com/post.php?id=7207003 ). These two games and additionally Pikmin 3 and Bayonetta make it worth, even without Smash and Kart. With the Wii-library to play too and upcoming Xenoblade Chronicles X and Zelda there are even more reasons.

Welcome, I went from an Xbox 360 to a Wii U.
Legend of Zelda TP is definatly worth it.
Metroid Prime series
There are a lot of good light gun type games like House of the Dead: Overkill and Resident Evil: Umbrella Chronicles.
The Conduit is a good albeit difficult shooter.
And lots more which more long time Nintendo fans might tell you more about.

I spent much of Easter Week playing Wii Fit Plus games! We had a great time on the ski jump, obstacle course. hula hoop, and many other games. We took turns trying to beat each other's high scores.

Not sure what else you get on WiiU Fit Plus, but it should be fun.

I also liked Endless Ocean: Blue World and Wii Sports' Resort.

This is a bit hard since in your OP, you cancelled out the 2 games I consider to be the pinnacle of local multiplayer gaming. But you really should get Mario Galaxy 2 (even if you don't like Mario games, you'll probably like this one. It's the best they've done) and Metroid Prime Trilogy. They're $20. That's $40 for about a 100 hours of gaming. You already said you liked Zelda so buy Skyward Sword (Wii, I don't think its on the eShop) and Windwaker. And if you have someone to play co-op with, then get Hyrule Warriors and Trine 2 (and/or Trine Enchanted Edition).
Most other games I could call are available, and arguably better, on other platforms. There are quite a few good indie games though. Stealth Inc 2 uses Miiverse well and I heard Affordable Space Adventures makes the best use of the Gamepad so far but I haven't played it. Then there's Devil's Third and Xenoblade Chronicles X coming out this year. I'm answering based on the taste you described in your OP.