Replacement Brifters for a shimano 105 2x10
The first thing to do is check the brifter out to see if its actually broken - maybe its just a cable broken at the brifter or inadequate cable tension or something. You can get your bike shop to check this out. If it is broken, you can look for people who specialize in refurbishing brifters (they do exist, but they are hard to find). 

As for replacement brifters, any Shimano 10 speed brifter will work on the right (you can easily find 10 speed Tiagra, 105, Ultegra, etc. shifters on the web (I see several at Chain Reaction Cycles, for example)). On the left, you just need a Shimano brifter set up for a double. You can buy just a left or just a right in some cases (possibly on ebay).

Other options are to use something like a Retroshift or bar-end shifters and a regular brake lever instead of a brifter. Microshift brifters are also Shimano compatible.