There isn't a spring on that float, I have a couple toros with that engine. Do make sure that when you put the float back on you have it right side up.
Well, at least I'm not crazy for not remembering seeing one when I took it apart the first time. I guess it's possible I had the float upside down. When I put it back together the second time (after thinking "oh well, maybe it got stuck. I'll just try it again without the spring.") I made sure I had the float pointing the right way. No leaks this time. It ran great and idled perfectly all the way down on low throttle, which it wouldn't do before.
I was confused about the spring because the Kawasaki service manual for that engine shows a spring and talks about making sure the spring is seated correctly on the float.
Guess I still need to find a few sets of extra carb gaskets for when I will inevitably need to replace them.
Thanks for the replies everyone.