Hi, catzor,
I have a 28-meg .pdf file of the Kawasaki Shop Manual for the rotary valve singles, that includes that model. I owned a G5 7 or 8 years ago. It had a spark system that, I swear, would spark through grease! PM or email me, and I'll send you the manual, if you want. kirkcheri (at) msn (dot) com
Meanwhile, I'll root around in it and see what troubleshooting help I can find. I just don't remember off the top of my head re: the battery, but I surely seem to recall no battery required at all for spark.
*EDIT* a quick read confirms - no battery required for spark. So, straightforward troubleshooting applies - you need a good spark plug (one that has been confirmed to spark well in some other vehicle); good connections in the system, including grounds; it has points and a condenser, so maybe check point gaps and give 'em a quick touch with a points file; check the resistance readings on the coil windings, although I don't know what values they should be, right off; and check the spark plug cap where it attaches to the the plug wire - make sure it's a good connection - maybe even remove the cap and "prune" the last 1/4" off the old lead to ensure a good connection.
Good luck!
K.