Thanks for the comments.

One thing I don't think I said was that I ensured the drill bit protruded from the Dremel only far enough to get about 2mm from the base of the hole. I had visions of wrecking the laptop if I accidentally drilled into some other component.

If I had to do this again I'd drill as far as I could, then I might try gluing a cocktail stick or thin nail into the hole in the plug with cyanoacrylate/CA/super-glue. I'd use the gel type so it didn't drip all over the place and make the problem worse though. 