Q. "I just bought a Hori VX-SA Kai (360) with a broken stick and I'm not used to changing parts yet so I wanted to make sure which sticks are compatible to put in and would appreciate the help."
A. The sky's the limit. You can see in this picture of the VX-SA Kai that your mount is a standard for JLF or Seimitsu or Hayabusa among the mainstream. You can keep the mounting plate and just get a new JLF and using this technique http://support.focusattack.com/entries/51042189-Joystick-installation-for-iCade-tough-jlf-clone-removal-of-gate-and-nuts. Try not to laugh too hard.
If you decide just to get a new Sanwa JLF you can order it from here: http://www.focusattack.com/sanwa-jlf-tp-8y-sk-joystick-no-plate-installed/ just save your screws and nuts from the original stick for the mounting plate. If you want to consider other sticks, here are just a couple more as example:
If you are going to continue with e Sanwa JLF there are additional things to upgrade if desired.