Now that the contacts are exposed, you can desolder them.

Make sure to allow ample time (at least 5-8 minutes or more) for the soldering iron to heat up. Otherwise, the solder won't melt.

Touch the hot iron to the contact points and gently pull on the wires to loosen them from the solder. They should release easily.

Make sure not to touch the contact points after desoldering them because they will remain very hot for a short while afterwards.

The speaker can now be removed. You may need to pull it with some force since there is a small bead of hot glue holding the speaker to its housing