Eh... Luigi's mansion? IDK man. I loved Luigi's mansion but it was too brief and not fleshed out enough to be # 2 of all Nintendo's creations... For me...

10. Paper Mario TTYD
9. Kid Icarus Uprising
8. Metroid Prime
7. Pokemon Red/Blue
6. Legend of Zelda... IDK Skyward Sword I guess.
5. Smash Melee
4. Super Mario 64
3. Super Mario Bros 3
2. Super Smash Bros Brawl
1. Super Mario Galaxy

I agree on Mario for Mario Kart (Double Dash & 7) & Sunshine & Galaxy (1 & 2) & Sticker Star and Pokémon for Emerald & Black/White (1 & 2) Mystery Dungeon Gates to Infiniti & Coliseum & Gale of Darkness & X/Y and Super Smash Bros. for Brawl & 3DS/U and Luigi's Mansion for Dark Moon

But you seem to be missing out on a lot of other good Nintendo IPs, like:

Dillon's Rolling Western 1 & 2

Pushmo 1 & 2 & 3

Spirit Camera: the Cursed Memior

Ketzal's Corridors

Kersploosh

Harmo Knight

Kid Icarus

Star Fox

Metroid

Fire Emblem

Bayonetta 2

Lego City Undercover 1 & prequel (3rd Party)

Wonderful 101 (3rd Party)

Rhythm Thief 1 & 2 (3rd Party)

Epic Mickey 1 & Power of illusion (3rd Party)

Bravely Default (3rd Party)

1. LoZ: Wind Waker
2. Super Smash Bros Brawl
3. Kid Icarus Uprising
4. Luigi's Mansion
5. Mario Kart Double Dash
6. LoZ: Ocarina Of Time
7. Pokemon Yellow (Its all about Pikachu!)
8. Super Smash Bros Melee
9. Mario Power Tennis (Is that the GC games name?)
10. Super Mario 3D Land

Honorable Mentions - Twilight Princess, Skyward Sword, Super Mario Galaxy, Super Mario Bros 3, Goldeneye, Star Fox 64 & All the other Pokemon games.

No Order

Metroid Prime
Super Mario Galaxy
LoZ: Wind Waker
LoZ: Ocarina of Time
LoZ: Link to the Past
Super Smash Bros. Melee
Super Mario World
Super Mario World 2: Yoshi's Island
Super Mario 64
Star Fox 64

Making a top10 nintendo games is a hard task. I loved DK64, Luigi's Mansion and MM, but they don't belong in the top 10imo.

For me, in no paticular order:
Super Mario
Link to the Past
Super Mario World
Metroid Prime
Ocarina of Time
Super Mario 64
Super Mario Galaxy
Pokémon Blue/Red (was rly close to put Gold/Silver instead but I went with Blue/Red since they are what started everything)
Donkey Kong Country
Super Metroid

Honorable mention to all Zelda games, Earthbound, Pokémon Gold/Silver, Goldeneye, SSBMelee, Super Mario Galaxy2, Paper Mario: Thousand Year door.

Paper Mario
Paper Mario The Thousand Year Door
Super Paper Mario
Fire Emblem Awakening
Super Mario Sunshine
Mario Kart DS
Mario Kart Wii
Mario Kart 7
Super Smash Bros. Brawl
Pokemon Heart Gold
Legend Of Zelda Majoras Mask

In no particular order (yes I know it is 11)

I can't say for sure what is the best, but these are my favorite Nintendo-published games.

1. Metroid Prime
2. Xenoblade Chronicles
3. Super Mario RPG: The Legend of the Seven Stars
4. The Legend of Zelda: Ocarina of Time
5. Super Mario Bros. 3
6. The Legend of Zelda: Link's Awakening
7. The Legend of Zelda: Skyward Sword
8. Super Mario 64
9. Pokemon Red/Blue
10. Advance Wars

Once I develop nostalgia for Fire Emblem: Awakening it should be high up there too. Also, Skyward Sword and Link's Awakening are practically tied. I enjoy them both for exactly the same reasons, oddly enough.

I guess I should make my top 10 (in no particular order):

Dillon's Rolling Western (my favorite Miyamoto game)

Mario Kart Double Dash

Pokémon Emerald

Star Fox 64 3D

Kid Icarus: Uprising

Fire Emblem: Awakening

Sakura Samurai: Art of the Sword

Ketzal's Corridors

Spirit Camera: the Cursed Memoir

Luigi's Mansion: Dark Moon

I've named 1st Party games only.

Savvy Style is another 1st Party IP, but its really a game for Fashion everything fans.

You said Nintendo exclusive games, but I am going to just stick with Nintendo published to limit the list some more. You only had Nintendo published games in your list anyway. I started with a list of over 20 games, and this is what I was able to cut down to. I tried to limit myself to one game in each franchise as best I could. I also tried to represent games across all the different genres that Nintendo dabbles in.

Diddy Kong Racing
Donkey Kong Country
Mario Kart Wii
Metroid Prime (Trilogy if you will allow it)
Perfect Dark
Pikmin 2
Super Mario Galaxy
Super Mario RPG: Legend of the Seven Stars
Super Mario World
Super Mario World 2: Yoshi's Island
Super Smash Bros. Brawl
The Legend of Zelda: Majora's Mask
The Legend of Zelda: Ocarina of Time

My Top 10 Nintendo published games:

10.) The Legend of Zelda: Twilight Princess
09.) Super Mario Galaxy
08.) Pokémon Blue
07.) Lylat Wars (Star Fox 64)
06.) Super Mario Bros. 3
05.) Mario Kart 64
04.) The Legend of Zelda: Link's Awakening
03.) Super Mario World 2: Yoshi's Island
02.) Super Mario 64
01.) The Legend of Zelda: Ocarina of Time