Rent a car and go around the island
You have got great suggestions already in prior posts. 5 days is also enough to rent a car and go around the island, which is what I did.
Akureyri (and surrounding areas) in the North is interesting and the drive around Eastern Iceland (hugging the coast) is also very nice.
Might as well check it out before they sell the island to the Chinese and then you may not be able to do it any longer!