Welcome to the club, Nick.
I do admire the XK, both in design and engine power, but being an old pensioner I am sticking to my very comfortable s Type.
I have not come across the SatNav disc ejecting itself, and can only suggest that you clean it with ordinary CD cleaner pad that you would use on CDs. Failing that, you could get a replacement disc --- a Satnav updated 2012 disc costs less than £30, and there are plenty around if you google "Jaguar Sat Nav update".
It is good to ge all the little things sorted out.
Regards,
Peter.