The rent for such an apartment in Reykjavik could be from 40,000 ISK per month if it's small and then upwards. I don't know if it's less expensive in Keflavik. I would have thought it was easier in summer. Students who come from the countryside rent apartments in winter and then often let them go in spring to go home to their town, live with their parents for free and work during their school holidays. Not that there is really any easy time of it.
Housing can be pretty hard to find over here.
Maybe it's easiest to take out an ad in the classifieds. Gives you a little more choice. You can find their office once you get here. The rent for such an apartment in Reykjavik could be from 40,000 ISK per month if it's small and then upwards. I don't know if it's less expensive in Keflavik.