www.zedder.com has some engine numbers as does buykawasaki.com under owner info/parts diagrams.
Get ahold of a 550 wiring diagram once you have figured out the year of the engine and you should be able to figure out how to wire the bike.
In general, you will need a regulator/rectifier, an ignition switch and a battery. You will need a starter button/kill switch and starter motor unless you plan on kicking the bike to start it.
Power will come from the alternator and go to the reg/rec. It will come out in on two circuits... the white/red which will go to your battery/main fuse and then to the ignition switch to become the source for several SWITCHED circuits and the brown wire which is the source for the unswitched circuits on the bike which are the ones that go through the two 10A fuses. By using the term switched, I refer to the igntion switch and not auxilliary switches that control the lights and stuff. The coils and starter button/kill switch are part of the switched circuits.
You will have to decide how fancy you want to get with the kart's wiring and then more specific advice as to how to wire it could be given... perhaps even a wiring diagram. If it were mine, I would likely use a switch of some sort to replace an ignition switch. I would run the main white wire from a fuse to it and come out with switched power. I would run this through a KILL SWITCH for safety. Then I would split the output from the kill switch with one leg going to the coils and one leg going to the starter button. This would be fairly easy to wire and provide positive shut off for the power to the coils and some safety in having a kill switch.