How many days is an excellent question! We only had a 17-hour layover, and next time I'd like to go back in spring (not too late to maybe see the Aurora Borealis) and spend two or three days. March is smack dab in the middle of Aurora season, I think - or at least, not after it is over.
There is actually quite a lot to do there (check this list: http://icelandtravelblog.net/2011/top-10-must-dos-in-iceland-from-a-locals-point-of-view/) and I know that I definitely want to drive the Golden Circle, soak in the Blue Lagoon, see a flock of puffins, go horseback riding, hike somewhere in the weirdly glorious countryside, and check out thermal springs/Gullfoss/Geysir.
We were there just a couple of weeks ago and it seemed like most of the standard bus tour options were up and running. There's winter stuff to do - snowmobile tours and whatnot - but yes, it also seemed like the tourist load was quite, quite low.
My recommendation for a place to stay is in Keflavik (short layover - we didn't want to spend an hour on the road to Reykjavik). We flew blind and got a place last minute via Booking.com and completely lucked out. The Svitan Guesthouse (http://www.svitan.is/) is welcoming, cozy, and run by a really NICE woman who knows how to make you feel at home. We got in after midnight and let ourselves in (nothing was locked). In the morning, we discovered the kitchen was stocked with eggs, bacon, fresh tomatoes, cheese, bread and crackers, cereal, milk, vegan "milk," butter, vegan butter substitute, jams and jellies, peanut butter, hazelnut spread, tea, coffee, and lots of wonderful skyr!
After breakfast we walked the three whole blocks to the waterfront and ogled the cliffs and the sea, then explored the teensy, quaint downtown of Keflavik.
Of course, if you elect to stay in Reykjavik, there are a lot more options. I just had to give a shout-out to the Svitan. :-)