If it is the stock non-vacuum powered petcock there is not much to it, see attached pic. I have read of others using something like toothpaste to lap in so to speak that tapered plastic item and the port it fits into. Do not know if that will help or not?
They are not hard to rebuild. Some attention to detail will get you smooth, leak-free, petcock. Here is a tutorial for rebuilding them. It is for Yamaha petcocks but the Kawasaki ones are similar.
How to rebuild vacuum operated petcocks
2016 Yamaha FJR1300ES
1982 GPz750 R1
1974 Kawasaki H1
1976 Kawasaki KZ400
1979 Yamaha XS650 cafe'
2001 Yamaha YZ426
1981 Honda XR200 stroked in an '89 CR125 chassis
1967 Triumph GT6
"If you didn't build it, it's not really yours"
it is indeed the non-vacuum version and the culprit is likely that little piece of conical plastic. I will try cleaning everything with my ultrasonic cleaner and hope for the best. Does anyone make that plastic piece, Z1 doesn’t seem to carry them.
1977 KZ650 C
Jets - Main 115 - Pilot 17.5 - Needles - Position 3 - NGK BR7ES
Give Pete a call. He makes some interesting stuff for the ZRX and a few other bikes. He might be interested in giving this a try if he has the time and if he could sell a few. I have bought from him several times and it has always been a great experience, and he likes to talk to enthusiasts.