The typical way of using a Shimano IGH with a dropbar is using a device called a Hubbub. It is essentially a little handlebar that plugs into the right hand side of the drops and gives you a mount for the usual twist shifter which comes with the IGH (This is what I would recommend). Another alternative is mounting it on the center of the bar with an extender like the Minoura space grip (though this is more annoying and less robust than the former option). JTek also makes a bar end shifter designed for drop bars as well. I have seen Walmart bikes with gripshifts in the center of dropbars, but these have bars that have essentially been cut in half, gripshifts installed and then bolted together according to the internet (google GMC Denali bicycle).
If you can look up (or measure) the cable pull for your IGH system, depending on if its roughly equally spaced, you may be able to rig a pulley system like a Problem Solvers Travel Agent to change the cable pull to work with the brifters okay. I think I read somewhere this works with some nexus hubs, but I could just be imagining. EDIT: http://www.flickr.com/photos/hiawathacyclery/2348490318/in/set-72157604007257815/ indicates some bike shop has managed to use a travel agent to get it to work: "That silver gizmo on the downtube shifter boss is a travel agent, intended to make short-pull brake levers work with v-brakes (aka linear pull). Coincidentally, it also amplifies the cable pull the correct amount to make an 8sp bar-end shifter index for the Alfine/Nexus 8sp hub. It works surprisingly well in the friction mode, too. ... It's as simple as that. However, I make no promises that this is a safe configuration or that it won't shorten the hub's lifespan. There is a small hesitation between gears 4 and 5, but it clicks into gear when the pressure on the pedal is relaxed slightly. Of course, anybody who's ever ridden an old 3-speed knows this already.... I should add that I believe the Alfine is a better choice than the Nexus. The reason is that the Alfine is more or less silent - any noise you hear is probably the result of a misplaced shift. ... Oh yeah, and we used an extra inline barrel adjuster on the outbound end of the travel agent. This is where the cable tension is adjusted to tune the shifting. The travel agent's barrel adjuster really just tensions the cable between the travel agent and the shifter." It looks like the travel agent mount is a custom one from Hiawatha Cyclery though (but shouldn't be hard to rig one up yourself).
Another thread on bicycles.SE (Is it possible to use road-style brifters with an internal gear hub?) says the Versa VRS-8 shifters are designed for this application, but I've never heard of them before. They also mention Sturmey-Archer hubs have brifters.
Switching to a derailleur and getting the wheel rebuilt may be possible, but you still have the expense of a derailleur, wheel rebuild, and possibly the purchase of an adaptor claw.
Also, not directly related: Installing drops changes the riding geometry of a bike - make sure you can still ride the bike with drops before buying something like a Hubbub since they're not cheap (though it shouldn't be hard to rig up something like that on your own).
Also, it seems your bike is fitted with V-brakes which have different cable pull than other brakes. You need to either get the Tektro RL540 (or equivalent) brake levers and use bar end / hubbub/ some other shifter mount, or use travel agents with STI shifters if you're able to get them to work.
Finally, some companies like Origin8 make bar ends with drops (http://www.origin-8.com/product-description/?prod_model_uid=1596). Not ideal, but might be something to look at.
