https://www.dndbeyond.com/subclasses/391719-way-of-the-dragon

Is this balanced?

No. I never even got past “Draconic Style” and the answer was “no.”

Draconic Style

When you hit a creature with an attack on your turn, but before you roll for damage, you can spend 1 ki point to add extra damage dice equal to double Martial Arts die to that attack, and to any other unarmed melee attack you make that turn. The damage type is determined by the type of dragon you chose to emulate.

Compare it to the 6th-level Kensei ability, Deft Strike:

Deft Strike. When you hit a target with a kensei weapon, you can spend 1 ki point to cause the weapon to deal extra damage to the target equal to your Martial Arts die. You can use this feature only once on each of your turns.