I believe any and all KLR650A and E carbs will work; there were some minor changes in 2008 and 2012 IIRC.
As far as the enricher nut goes; a cracked/broken one can cause a lean condition. You can buy a replacement at Artic-Cat (PN 5507-025) for cheap. Alternatively, Stead engineering makes a brass replacement that will outlast the bike.
Beyond that, make sure the carb is clean (when you think it's clean, clean it again!) and that the jetting is right. Stock jetting will work OK as long as everything else is stock, otherwise pick up a KLX jet kit from Eaglemike and the bike will run better.