Army, you probably have a better book than I have. Mine is so old that the pages are brown and crumble. I'm not sure if it takes more ingenuity to open the book than to repair the part.
I searched quite hard to find a proper replacement brake cable, but the part # for a '57 seems to be unique. And not available. So I'm not sure I should go up to a 544 brake cable or down to an older model PV444. I pulled some measurements off the old cable and will take a photo.
I was unable to weld the sheath back to the end fitting because the sheath was made of some sort of plastic. I was able to get the end fitting to slide back over the sheathing, but the overlap was maybe 3/8" at most. I used an epoxy glue on the interface. Then I made an exterior "cast" out of fiberglass overlaying the 3/8" joint and extending an inch or so down the sheath. It seems strong enough just looking at it, but I suppose a high risk of failure on first use.
For now, I think I'm going to try to get the brake cable hooked up and see if it works.