I am never sure as to what the right greeting should be for many of the Jewish holidays, BUT I know that when it comes to Chanukah “happy” is the right word.
So, Happy Chanukah/Happy Hanukkah. Enjoy.
As for the Jesus angle. What can I say. I am sure that over 50% of Americans, if asked, will say that Jesus celebrated Christmas. About 90% of Americans think he spoke English and about the same think that he was a Christian.