I bought a 1998 XJR about 2 weeks ago that looked and ran perfectly, until I got it home. The next day I went to start it in the morning and it cranked but wouldn't start. I bought and installed a new battery after testing the old one and finding it questionable. I tried starting it a few times and after about 5 minutes it just fired up and was fine. Took it to get gas and after being parked for 10 minutes it started fine, took it to the mall and after 2 hours of being parked it started gave me a little grief. I have done reading, testing, swapped, bought, installed relays, cleaned terminals, you name it. Then I started to realize that I could tell when it would start perfectly because all the lights on the gauge cluster illuminated when the key was turned, but when the key was turned and only the odometer, gauges, signal indicators, and headlight displays illuminated it would only crank and throw warnings such as "electrical failure" "suspension failure" restricted performance" and occasionally "wiper fluid low". It took longer to start the longer it sat, until I did some research. I read that the XJR uses a Mercedes transmission and that the dual linear switch can fail causing cranking but no starting and I read that one member replaced a faulty circuit board under the shift bezel after having similar simptoms and that fixed his car. I have it down to a science now that I can start it by keying on the car (i don't have to turn it over) and if all gauge cluster lights don't light up I key it off, slightly jiggle or move the shift knob, and try again. Normally between 5-10 attempts I find the shifter's "sweet spot", all lights illuminate, and it starts perfect and off I go. My landlord was a jag service tech twelve years ago and remembers a laughable problem with these symptoms that is XJR specific, was about a 15 minute fix, and said the linear switch under the shift bezel sounds like something he remembers. Recap of what I know:

1) good battery
2) all fuses good
3) cranks and doesn't start when only a few gauge lights illuminate when keyed
4) starts fine when all lights on gauge cluster illuminate when keyed
5) obviously electrical
6) moving shifter expedites all gauge lights illuminating
7) takes longer (more key/shift attempts) the longer it sits
8) regardless of which lights light up when keyed, the radio, starter, seats, windows, keyless entry, pretty much everything will work: EXCEPT THE A/C CONTROLS.
9) There is a small 6" wire with a female blade connecter coming from the wiring harness near the battery that is unplugged and I cant tell where it goes.
10) There is 3 10" wires surrounded by a wire loom that attach to a circular tube-shaped 1" long grey plastic connector that is disconnected and sitting near the intake hose and the fender, coming from underneath the box (that holds the computer and has a sensless plastic hatch on top) under the hood on the passenger's side).
11) I can take pictures and have a digital camera, but have never figured out how to post them on a forum before.

I sold my classic muscle car that I had restored myself because I wanted a solid, comfortable, fast car with a/c and a radio: burning up in my hotrod stuck in traffic so that I couldn't even enjoy the 550HP motor got old after a few decades. While mechanically inclined, I am not electronically savvy, so any help that you can provide would be greatly appreciated. Thanks.

 

It may have entirely nothing to do with the shifter, played around with just the key and after about 20 times of keying on and off, all lights lit and a/c controls illuminated, and it started just fine. Key switch? Anything?

 

It sounds like the shifter to me, there's two large nuts that adjust the cable, if they're loose or out of alignment (in relation to the switch)by a tiny amount it'll throw all sorts of messages, "the scroll of faults" so to speak, all warnings like you experienced. It's on your passenger side. Look for tutorials on here about removal of the centre console and check it out.
If you feel it's the key/lock barrel, then inspect under the steering column, remove the lower plastic shroud with a magnetic phillips screwdriver (x4 screws) and check the wiring connections. The ignition switchpack clips right onto the back of the steering lock that's where you should start checking wiring.
The mystery wiring, I suggest opening a photobucket account and upload the pictures to there, then post the links of the pictures here so we can see what you're on about.

 

Does this plastic component just pop off? Are there hidden screws, plastic tabs, is it glued, I would hate to break or scratch something...

 

I no longer believe that moving the shifter helps, it takes anywhere from 1-20 key attempts for all the dash lights to light and it to start. it will crank regardless. Help? Anyone?

 

This sounds like main power supply to the engine/front end harness - check the fuse box connections under the bonnet, undo the nuts connecting the thick black cables to each fuse box and clean them with wirewool/brush and reconnect. Also look at the other side for a similar thick black cable and do the same thing, it should sort it out.