Have you swapped the Starter Relay over with an identical one?
May be the Ignition Switch or Actuator Pin Assembly which is a big problem on the KJs...don't know if they changed this Pin Assembly on the KKs
You have changed the ignition switch I see.
On the KJs the switch position on the steering column is held on by a single Torx screw.
I had the problem and some other KJ owners have had the problem in that the switch when mounted can be rotated slightly advanced or ******** with respect to the Actuator Pin Assembly.
This either has the effect that the Switch is only just making contact when rotated to the Start position or in my case with a Manual box...the Jeep would start when I depressed the clutch...instead of waiting for Ignition Switch to also be activated.
The cure for this is to slightly elongate the hole that the Torx Screw goes through so as to slightly advance/****** the switch position!
While working in that area check the connector on the Switch is on firmly!
