This is my first post on the forum as I've just joined LCC and it's probably too late to be of any help to you.
I had a similar bike for many years until I carried out some long overdue maintaince on the hub and then innards crunched horribly and came through the hub shell NO MORE NEXUS! The hub was kapoot anyway and I was simply trying to get another few hundred miles out of it to make the decision on a new bike easier and less pressing. Any road I hope I'm not teaching you how to suck eggs but by far the best bet is actually not to remove your wheel. If you don't know where the puncture is get your bike upside down and simply pop your tyre off on one side and pull your inner tube out. This gives you access to most of your inner, just not the bit stuck behind the axle, and you should be able to find the puncture fairly easily using the normal route of pumping the tube up and listening.
I nearly always mananged to spot when the puncture first occurrred and jump off the bike and count the spokes back or forward from something significant on the rim (sticker or valve etc) and then I only I had to expose the inner on that area.
I really like internal gears and brakes for commuting, they make so much sense. I always thought that my hub brakes worked great and would gladly go back to them if it was easy but not many manufacturers seem to be using them as they're not sexy.
Cheers and good luck.