I went for a bachelor party two years ago so not quite going with your wife, but here are some recommendations.
- Go to the Blue Lagoon for a thermal spring, its a very unique experience.
- Do the brewery tour, I think there is only one, excellent beer there.
- Eat a hot dog from the famous hot dog place
- Try puffin and whale
- Do the free walking tour
- Pre arrange transportation from the airport, it is very far from Reykjavik
- Nightlife starts late, it's all on the same street whose name I forget.
Maybe you know this, but Icelandair offers a free layover in Reykjavik...this means that you can fly from JFK to Reykjavik for a few days and onto Europe at no extra charge, and back to the US. Well worth it to combine with a European city.