When you say you have installed a new Ignition switch do you mean the actual electrical switch or the Actuator Pin Assembly which rotates the switch when you turn the key to start?
I do not know the KK at all but the ignition key system should be similar to the KJ. The KJ uses an "Actuator Pin Assembly" to rotate the ignition switch. This consists of a hollow square tube made from some cast metal, the hollow section has a long plastic sliver protruding from the end of this hollow tube....this sliver extends when the key is inserted to press on a separate switch inside the main ignition switch.....this lets the vehicle "chime" at you if the ignition key is left inserted while the driver's door is opened.
This is not important but it does weaken the structure of the square "Actuator Pin Assembly" which slowly bends until it breaks.
Weakened or broken "actuator Pin Assemblies" are a common problem on KJs.. and Dealers only sell the complete steering column assembly but Dorman parts are available.
On the KJs this "Actuator Pin Assembly" is mounted onto the Steering column by a single Security Torx Rotational mounting device ie. you need a Male Torx Driver with a hole devised down its center.
I have seen some cases where the rotational positioning of the "Actuator Pin Assembly" with respect to the actual Ignition switch is slightly advanced or not advanced.
So the hole on the "actuator Pin Assembly" through which the mounting Security rotational positioning device goes through needs to be elongated slightly so that the "Actuator Pin Assembly" can be advanced or not advanced relative to the ignition switch before firmly tightening the mounting rotational positioning device.
So you may have to inspect your "Actuator Pin Assembly" to check if it is starting to crumble and if it is advanced enough so that it moves the Ignition Switch all the way.
On the KJ you can leave the ignition switch hanging freely, place key in the ignition slot to enable SKEES and unlock steering lock. Then use a small flat screwdriver to rotate the ignition switch to start the engine. This will prove if there is a fault in this area or if your poor starting is something completely elsewhere! I ran like this for a few days on my 2002 Export Diesel until I was sure that I just needed to advance the Actuator Pin Assembly Slightly!
Otherwise you may need to have Codes pulled out to see what the Jeep is complaining about or check all grounding points are clean and tight and that the Park/Neutral Interlocks are functioning properly!
I am getting so ****** off by this Mother Grundy foul language checker!!!!
I hope all you know what a Rotational Positioning Device is!..you need a Torx Driver to rotate it!