what plinkie said, when u trigger the animation you trigger the damage also towards the enemy if the enemy doesn't have a special status like "defence mode" or to far away...
instead of overlap tough i would use on collision if u do the damage swing on contact, if you want to have multiple parts of the enemy where you can do more damage... than the arm for example you either need a ragdoll character and set different dmg multipliers for head, arm, torso, legs etc... or use image points and hidden sprites on the character and when the sword hits those you can compare by length of sword over the image point to pick farthest point or closest to the handle and give it more DMG, like a critical hit.
but for simplicity wise, on key pressed play animation + do dmg to enemy.