Hold key in bench vise.
Place slotted wedge around backcheck wire. This will align the tool to pull the wire out of the wood without tearout.
Place the hammer extractor under the backcheck and on top of the wedge.
Run the drill to spread the extractor. This will pull off the backcheck. If you're lucky the wire will come out too.
If the wire is still in the key, clamp the vise grips on the wire so there is enough clearance for the extractor to fit under it and on top if the wedge.
Reset the extractor and run the drill to pull out the wire.
Wire is out. The hole is undamaged.
This is how it is done.