Here is what I found. There are several hostels, but the best one for location and atmosphere is IH located 1 block from the best beach. A block back from there is the Apart Cavancha Iquique which offers 1&2 bedrrom apartments. They are very nice with complete kitchens, wi-fi and cable tv. I stayed here and really enjoyed it. To stay at these apartments, contact Fabrizio by e-mail: firstname.lastname@example.org
They are priced quite reasonably and fabrizio is very helpful.