Run around the edge of the room, keeping your distance from Stagnox while circling it. If you get too close, it will swipe at you. Eventually, Stagnox will do a horn attack that is very slow, allowing you to run over to Stagnox's backside. Use the whirlwind to blow away the purple fog on its rear, and then slash away at it with your sword.

Eventually, Stagnox will recover, so run away again and start circling once more. Repeat the process of slashing at Stagnox's rear, and eventually it will fly into the air.

Stagnox, while hovering over the platform, will drop three blastworms. After this, Stagnox will eventually fly down toward you. You will know that it is about to fly down if you can see some wind blowing across its horn. Hit a blastworm with your sword and then use the Whirlwind to blow the blastworm at Stagnox when it is flying across the platform.

After the explosion, he will drop to the platform, where you can hit his backside while he is stunned. Repeat the process until you have defeated Stagnox.

When you have defeated Stagnox, open the treasure chest to get a heart container, then step into the blue light. Now you can go back to the Tower of Spirits to get the Snow Rail Map.