Set up drill press with jig and backcheck holder..
Place key in jig.
Place new backcheck in holder and align wire with hole in key. Make sure wire is vertical and not angled - wire will bend if not aligned properly.
Press wire into key. Stop pressing when the backcheck is at the correct height. Set the drill press depth stop to this depth.
Release the press.
Here is the procedure in action.
Finished installation. If the wire is loose in the hole. Inject thin CA glue at the base of the wire to tighten the fit of the wire in the wood.