The ignition amplifier looks very much like one from a stereo, an unpainted aluminium box shape with cooling fins mounted on the top. These were always prone to breaking down if they got too hot, the only recourse being to replace the unit, no repair is possible. On early V-12s the amplifier was mounted right in the middle of the 'V' created by the two banks of cylinders, and on the later cars, of which the '89 should be, it should be placed on the top of the panel going across the very front of the engine bay, which would be the bonnet shut panel if the Jag bonnets opened normally! This applies to the saloons, although I don't see why the KJ-S should be any different.
A very good source of information is fellow forum member Diggs, who is a master Jaguar tech. He should be able to give you an idea if the amplifier should check out okay.