Bring the gem and dragon breastplate to the smith Grisha in the dwarven halls, and he can forge them together into a potentially better breastplate. If the new breastplate isn't very good, you can buy back the original, and just use the forged version to get the dragon helm from the king and wear the full dragon set once (for experience points and a level of a skill).
You can quicksave after he gives you the new armour, but before opening your inventory and identifying, then quickload to get the best stats (move the armour and save if you want to keep it but still reload to try for something better).