I found a site a while ago called Velosolo, the guy who runs it seems to be quite knowledgeable about conversions, and he has a couple of kits on there - you might be able to get away with ripping your cassette off and replacing it with a cog and a bunch of spacers.
Because of the vertical dropouts you'd still need to tension the chain somehow (the obvious way being a tensioner).
That approach would certainly get you to a "fixed" state (an would allow you to change cog size on a whim), you might need to look a little more closely if you want to mix a hub in there instead, but it is definitely worth asking them the question - I've contacted this chap before and he's very helpful.
You might have to settle for less-than-perfect chain line, but at the end of the day the chain you're running right now is designed to traverse a cassette, so that's not going to be a showstopper. As regards chains, 1/8" is certainly the convention going back in time, but most fixed cogs & chainrings will also come in 3/32" variants, you should have no problems there.
If you do find something on the web, you'll first need to remove your existing cassette. To do this, you'll need an appropriate lockring nut and either a chain whip or a vice. Park Tools have a selection of nuts that fit just about everything that moves. If the cassette is either Shimano or Sram, you'll need one nut (FR-5, but double-check), if it is Campy you'll need a different one (FR-11, but double-check). In my experience, its a lot, lot easier to stick the cassette into a vice than to use a chain whip. As a tip, a Black and Decker workmate makes a great vice!
Your other alternative for removal is to take the wheel to your LBS and ask nicely. For someone with the right tools it really is just a 30-second job, you might find if you ask them nicely they'll loosen everything there and then, and not charge you... But if you did this, convention is just to buy something (anything) from them as a token of thanks.