If your XK-8 will do everything in the morning except turn the starter, you may have a defective starter relay. These are easy to test if you can find the thing. After extensive searching on the innertubes,including this forum and a lucky guess, I found it.

Guess what; It wasn't where anyone said it would be. See attached picture.

The shot is the passenger side of a U.S. car, at the back of the engine bay. Note the three rectangular plastic covers, one of which has been removed. Note the white vacant relay socket. I pulled the relay, so you would normally see a small brown cube there. That's it!

Pull the relay and test it as follows. There are 4 connectors. They are internally connected in pairs across from one another. Attach a meter or lamp across the two connectors that are perpendicular to each other. Apply 12VDC to the 2 connectors that are parallel to each other. You should hear a soft click when you apply the voltage. You should also see or hear evidence of the relay actually making contact from the meter or lamp. My relay showed this about half the time which suggested I needed a replacement relay. Jag, Bosch, and Autolite in descending order of price all make these. Bosch is highly recommended although Autolite is probably just as good and is almost universally available.

Hope this helps.