Iceland June/July 2016: I didn't know where to leave my bike box during the trip around the isle. My flight was late and I did not want to start up at midnight. Therefore I checked in at the Ocean View Guesthouse in Keflavik (about 10minutes with a taxi).
There I could finish my bike on the next morning in a heated Garage where I could leave the box for the next 10 days without booking a further night at the day of departure. The People were quite friendly and did not demand additional Money for the storage. You can expect prices starting from about 70Euros in main season for a simple room with breakfast. I think this could be interesting for you!