You can't really ask someone to give you a nickname. Real nicknames just happen
In any case, Kate is a pretty short name so I don't think you really need a nickname for it.
Know what I mean? My name is Nicoletta, but since that's long people just call me Nicki because it's quick & easy. Beside, isn't Kate already short for something? Like Kaitlyn or Katherine or something?
You don't get to chose your own nicknames. They come from something that happened. I shoud know, I have had a lot of nicknames in the past...(Piglet, gnat, picoonat, Cartman, random, spaz ect.)
^Just take everyone's advice. You can, er, choose a nickname and ask people to call you that. That's what I do. My name is Katie-Ashlyn and people call me Katie, but I got Kate as a nickname.