There's some great suggestions in the other responses already. Some of these will probably overlap.
I love Kai to bits and have wondered about a longer name with Kai as a nickname too. The ones I like:
Makai
Makaio
Micaiah (although I actually like Makaia better and for a girl)
Malakai/Malachi
Caio/Kaio
Cairo/Kairo
And then there's the more biblical ones:
Hezekiah
Hizkiah
Kenaniah
Mordecai
Zedekiah
Zachariah
Zechariah
and I guess Micaiah belongs on this list too.
Plus a couple more:
Caius
Caiaphas
Kaiser
Kairos
Like any?