Reykjavík offers a wide range of accommodation possibilities, for all budgets and tastes.
Iceland is known for its very high-level service standards, and you will find that most places, regardless of rate, are impeccable and professionally staffed.

Although you can walk to/from most places, Reykjavík has an excellent public transportation network, at accessible costs, which makes it easy to plan your trip to just about anywhere:
In addition to the recommendations below, the city offers many good deals on small apartments to rent for a few days. A simple search is bound to cover all your needs.