Make sure you are in the reconstruct tab rather than the transmute tab. Then find the piece according to where it drops (mythics are in the bottom section). As long as you had that piece bound and thus collected at some point after the stickerbook was introduced, it should allow you to proceed.
It will also require between 25 and 50 transmute crystals, if you don't have enough, it may stop you. At the next screen you'll choose a trait from those that your character has researched, and the item's quality provided you have mats. Then you'll be given a confirmation screen and be able to make it.
The only other problem I can think is that you can't reconstruct crafted sets, because you can just craft them again with a different trait at the cost of mats rather than transmutes.
If you're still having issues, could you let us know what set and piece isn't working for you?