You could certainly use a soldering iron or other heat to remove the contacts from the sliding piece. Then take some thin PCB material, cut it to size, cut away copper so that the pattern matches the bottom piece. You would be advised to put epoxy in the grooves where copper is cut out and then sand it flat so the contacts glide smoothly. If possible, I would probably cut the bottom piece out of delrin or something like this and embed the little strip of PCB in it, but it is unclear if there is sufficient room for this. If you broke the contacts, you should be able to find comparable contacts inside an old slide type switch.
This would be a couple hours of work, you would really have to love this walkman to do it. Like the way you love an old coffee cup that you continue to use after dropping, breaking to pieces and then glueing it back together. For the second time