I looked at the International Booking Network site for Hosteling International, which I frequently use, and they list 1600 krona for a dorm bed ($US22) or 6400 krona for a 2 bed room ($87). You generally need a membership (you might be able to pay a surcharge for a single visit, but need to verify that with the hostel directly).

From its web site:

Reykjavik Hostel is perfectly located in the City, in the Laugardalur area . LAUGAR, the largest swimming and spa centre in Reykjavik is just adjacent to the Hostel and all the biggest Sport's facilities in Reykjavik are within walking distance. Close by there are supermarkets, banks, restaurants and a pharmacy. It is possible to purchase a number of tours at the hostel, both long and short. For most trips, there is a complementary pick-up service at the hostel. The staff at the hostel also provide information on other youth hostels in Iceland as well as assistance with booking.
The Flybus from International Airport in Keflavik brings you upon request to the hostel in about an hour. Cost 1100 ikr ($US15).

Here's the booking site and you can bring up the Reykjavik hostel quite easily:
http://www.yhabooking.com/ibnpub/english/index.asp