There are three ways to trace a car like this: Through a Registry, through the genealogy of the car or the genealogy of the owners. The last two are going to be difficult.
There must be an XK registry. Go to the JCNA website and they have a listing of Registries (only if it is associated with JCNA). Check there. Then go a Google search for Registries. That is the easiest way and you MAY get lucky.
Since you have the VIN of the car, you have a remote chance of going to DMV to see if the car is still in the State. This is a slim way to go since the records are extremely old and pre-computer to say nothing of pre-standard VIN numbers. Most DMVs require proof of ownership to do this but you may be able to slide by. You are going to have a hard time with the dealership as well. According to a quick GOOGLE search, the original Koster-Swope dealership burned down on May 3, 1961. Chances are the records of this 1960 XK150 were destroyed as well. Maybe not. If not, they could have been discarded when the dealership was sold. So the first step is to contact the original dealership in Louisville.
The other method is to track down the owner and/or his family. This to may be difficult. If you have the original Bill of Sale, you have the original owner's name and address. Start an Internet search and see if the Owner's family is still around. In my own case, I found the daughter of the original owner of my '67 who loved talking about her deceased father and the saloon she rode in as a child. She sent me a photo of the car new with her father standing next to it, but had no idea how or who he disposed the car to. UK records were destroyed prior to 1974.
Good Luck
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