Wow, that is an amazing painting. It makes my heart smile.

(Sorry that I'm really corny.

)
Hmm, I wish there was an easy way to ensure Conan would see it. I know a guy who did a painting for
conanvsbear.com, photoshopped his email address onto it, and was subsequently contacted by someone from the Late Night staff requesting a poster-sized version to give to Conan. Conan then had it framed and hung it in his office. You can see it in this pic on the wall behind Max:

I would suggest mailing a picture of your painting to one of Conan's assistants: Todd Blass or Laurie Scollar, c/o Late Night, 30 Rockefeller Plaza, New York, NY 10112.
Good luck.