Desoldering can be tricky and often requires special tools. Pinch the broken wire in a small pair of vice grips or any type of clamp that you wish to use. Heat the broken wire while in the vice grips. With the vice grips also hot it won't suck as much heat out of the wire. As soon as the broken wire gets hot enough to melt the solder, pull the plastic connector off of the broken wire. Reinserting the new wire will be similar process. Clamp the new wire with Flux on it and heat it while its in contact with the hole in connector.
Other options include using a desoldering wick to pull some solder out of the connector. Not all solder, and not all methods are equal. If you used lead-free solder, that certainly requires additional heat. That connector probably would benefit from a quality lower temp solder.
I know it looks like a mess but I think it's still salvageable. Another trick if you find yourself out of hands would be to have the broken wire clamped in small vice-grip and elevated while the connector is tied to a small weighted object, maybe a decent size bolt, pulling it down. As soon as solder begins to melt off of the broken wire, the weight will pull the connector off on its own without requiring you to grow a 3rd arm. You might have to get creative. Good luck.
Elon