You'll need to rebuild the circuit but its not a difficult one. Just look for the diagram for your controller, its under Xbox one controllers/rapid fire.
What you did wrong is clearly you used too much heat and too much time for the solder points. Microsoft uses a very hard tin which needs a lot more heat than regular soldering tin, next time add a little of your own tin, then try to mix it with the point of the soldering iron. Crank the heat up a lot and don't use like more than 2 seconds for each pad.
When you have removed the piece you can then turn down the heat just enough for your solder to melt so it will be easier on the pads.
Another thing that works well with Elite controllers is destroying the damaged stick so you can remove pin by pin and not worrying about removing the solder with a suction tool.