After the garage had a look at it, they decided they want to replace the whole
distributor at a cost of around $500 + labour.
I just decided to have another look at it and see if I could locate the ignition module. After a few minutes, worked out that there was a metal cover on the side that could be removed with 2 screws. Inside is what looks to me like the ignition module images I saw on google. This black plastic piece is completely surrounded by thermal grease which seems to have leaked out of the black module.
Does that sound like the location of the problem? Should there be any grease at all outside that black plastic thing? There was at least a desert spoon's worth of grease around it.