Logically speaking, there is no such word as "kate" so certainly there's no Korean word for it. Moreover, kate is mostly used as a name.
So Kate converted as a Korean name is usually "keiteu" (케이트) or it may also be as "gade" or "kade" (카데).
Note: "g" and "k" (ㄱ) in Korean are mixed (or similarly pronounced) which means they'll have the same character. But here we used k' and g' which have a sharper tone whose character would be ㅋ.