Switching temporarily back to the slides...Wurlitzer Used bakeilite slides, they are very fragile. I had one that broke. I was able to fix it by laying it on a flat surface with a piece of plastic cut from a baggie under the repair area. I taped everything down so it was lined up and used "gap filling" cyanoacrylate glue on the break. Let it harden for half an hr than untaped and carefully peeled the baggie off. I used a file with the part on the table to remove burrs and high spots.
Good improvisation- with due caution you can actually do a lot with superglue. (Cuts both ways of course; a lot of good or a lot of bad!)
I took a thin strip of credit card and sanded it down so it was thin enough to slide smoothly without undue bulging and used it under the stub of the existing slide and under the extension (also a chopped up bit of credit card stacked four layers high) with the catchment for the register mechanism arm on top of it. It acted as a joining underlay twixt the two where they abutted each other end to end. Cyanoacrylate adhesives do really well on that type of plastic.