First, use the Double Clawshots and latch onto its tail when it is not attacking you. Immediately after latching onto its tail, use the Iron Boots to bring it down so its armor breaks. After doing this a few more times, he will rip off the rest of the armor and it will start to rain. The rain will bring floating plants from the ground that circle the battleground. Use the Clawshot on the plants and circle around the dragon while dodging his flame attacks. When you are behind him, use the Clawshot on his back and attack the jewel with your sword. After one or two more times, he will be defeated and you will get the last shard of the Twilight Mirror.
