Regulator/Rectifier has nothing to do with spark, nor the main part of the generator. Also, I would not attempt to remove the wire from the coil. It may or may not be removable from the coil. Kawasaki does not offer it separately, so it is probably not replaceable. If it's made onto the coil, you will be buying a new coil if you mess up the wire.
First, I would suggest you replace the plug cap. It's an NGK LD05F. It has a resistor in it and should have 5K ohm resistance. You should be able to go buy a new one at your local dealer for less than $5.00. We sell that cap for $3.99. The cap screws onto the wire. There's a "sheet metal" looking screw inside it where the wire goes.
Here are the electrical specs for your machine:
Here's the ignition system diagram:
You need to make sure you don't have an issue with the ignition switch, kill switch, or wiring. If there is no issue there, I'm going to say you have a bad pulsing coil or a bad ignitor, or "CDI". We've never found a reliable test for the ignitor. The best thing is to find one from a friends machine that is working and switch them. The one for your machine fits 1996-2011 KLF220's and KLF250's. They are VERY expensive. You can buy one at amsdiscountparts.com for $328.51, part number is 21119-1457, but of course they are non-returnable, so you need to try someone else's before buying one. The pick-up coil, or pulsing coil, part number 59026-1135, sells for $101.46, again, non-returnable.