Accommodation is always harder to find in the summer, as lots of people switch their offers to nightly rentals for tourists instead of longer-term leases. My advice would be to ask around and see if the designer you'll be working with knows of anything available. A lot of people here do seem to find flats through personal communications.
Otherwise, if you're stuck looking on your own, there are some websites that you can use.
leiga.is/leit/default.aspx?search=1
mbl.is/leiga/
leigulistinn.is/
The first two are free; the last one costs about 3600 kr for one month of access. However, as a non-Icelandic speaker, it's also the one I found to be the easiest to navigate and use, and it's generally the one I had the most luck with.