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Overall the 2001 XK is a solid vehicle but did have some issues that you should be concerned about. The one concern I would have is the engine. On the early V8 engines they had issues with timing chain tensioners. The plastic tensioners can fail causing the timing chain to jump and even break. You need to listen to any abnormal engine noise on a cold start. The cam covers need to be removed to inspect this. The cost for a complete chain job could be between $2500-$3500. Next pay close attention to how the transmission is shifting as the 1997-2002 had some trans problems. Some other costly high mileage failures are the ABS control module and the electronic throttle units causing fault warnings to come on. Also since this vehicle is a convertible the rear top lift pistons and hydraulics lines in the front develop leaks. I know this sounds like a lot but it really all depends on how the vehicle was maintained and the mileage on it. If you have any other questions or concerns please let me know.
So, if the 13,000 miles is correct, what's the possibilities of engine or transmission problems? Also, is there an issue with the low mileage - the car not being driven regularly?Thanks,