Hi there,
thats an adventurous destination for your honeymoon!
I went there in November and really enjoyed it - amazing scenery - I havn't been any where else where in one day I saw geysers, climed volcanos, crossed frozen waterfalls and saw the aurora borealis.
Reijkavich is very small (the airport is about 5 mins from the centre of the city!), and I found it very expensive! E.g. a pizza for lunch in a very normal restuarant was 20E, a bottle of house wine 30-40E, glass of wine 10E.
It could be worth doing a jeep trip all around the island, and maybe going ice-diving if that's your thing.
Its a nice destination, but I cant imagine spending more than a week/10 days there - you could maybe combine it with a trip to Greenland where the snow safaris look amazing fun.