Don't feel like looking it up, but who is Longclaw?
Darkness lay over the world and a hero, Azor Ahai, was chosen to fight against it. To fight the darkness, Azor Ahai needed to forge a hero's sword. He labored for thirty days and thirty nights until it was done. However, when he went to temper it in water, the sword broke. He was not one to give up easily, so he started over. The second time he took fifty days and fifty nights to make the sword, even better than the first. To temper it this time, he captured a lion and drove the sword into its heart, but once more the steel shattered. The third time, with a heavy heart, for he knew before hand what he must do to finish the blade, he worked for a hundred days and nights until it was finished. This time, he called for his wife, Nissa Nissa, and asked her to bare her breast. He drove his sword into her breast, her soul combining with the steel of the sword, creating Lightbringer, while her cry of anguish and ecstasy left a crack across the face of the moon
If Jon's sword is already Lightbringer it doesn't make much sense to have to kill Dany to make it what it already is. However, maybe Jon has to create a new one. Following what happened before, maybe killing Dany with it will make it the "new" Lightbringer.
Just seems convoluted if Lightbringer already exists. It's either Jon's Sword and he needs to learn to use it or it's out there somewhere waiting to be found
The Prophecy says he will pull it from fire and it will help defeat the Knight King. The sword was said to glow and radiate heat, burning whoever touches it.