Well, it's the moon. What do I really say to that? xD
Here's a semi-spoiler, semi describing sentence about Snow's version of Kage: He only exists in Snow because he wants Leon to be happy. That's it. He literally has no other motivation other than, "I want to ease his suffering"... At first, anyways. This sentiment later extends to the rest of Kage's closer friends, like Haruko, Kuro and Nezu. He otherwise works the job he does (ie, assassin) because he needs money to support himself and Leon (who, if you checked chapter one of Snow out, was altered). Since Leon spends a lot of time taking care of Kage even before that Dual Ice Age bit, Kage develops a great sense of gratitude and figures repayment is necessary.
In all honesty, the relationship between Kage and Leon is less friendship, more "A Man and his Dog" sorta thing. Leon was the first human to treat Kage in any many of kindness, and stays sincere in their friendship. Kage became extremely loyal to him as a result of this kindness (as well as other things; I'm pretty sure I mentioned it in Kage's Age Meme thingy).
So yeah, tl;dr: Kage is basically a loyal kit to Leon, his human-turned-demon friend, and has a reason to live via lessening his friend's suffering.
After all, Leon's life was changed entirely by turning into a demon. Due to a stigma in human societies against demons, he essentially lost his whole life once he changed. His previous job (which was a pre-school/elementary school teacher/caretaker), he couldn't return to (and a love interest of his worked there, so he didn't see her again). In addition, since he turned into a deadly-looking demon... Yeah, his entire life basically got shot down thanks to a freak blizzard.
Even if he rarely shows this stress in Snow, Kage can read auras (albeit faintly). He's better at reading the environment, but can sense enough to know how Leon feels on the whole thing.
So yeah. Lots of words for a moon, huh? xD