Where can I get a new nose wheel centering cable, or get my old one repaired?
Search strings: steering cable, nose cable, nose gear cable, centering spring
McFarlane’s, in Vineland Valley, KS, has been recommended; their link is in the BAC Links section.
Both Aircraft Spruce and Wag Aero can also ‘fix it’. To make that happen, you MUST NOT call them and just ask for the part for a certified plane. They do not have a replacement FAA-PMA cable available. They also cannot officially repair certified parts, as they don’t have an FAA repair station license for it. What you must do is call them and tell them you have a cable that you need repaired, and ask whether you can send it in to them for repair. I doubt that they will ask, but if they do, say it is for a home project. They can then transfer the clamping bolts, spacers, and attachment clevis eyes onto a new length of cable, and roll-swage new single-shank ball-ends on the cable. It is relatively inexpensive.
You don’t want to go to Beech for a new fixed-gear cable. For a pricing perspective, Beech has plenty of these for $438 each (fixed-gear). The retract cables are more reasonable, at $65 each, but these seldom go bad. The high fixed-gear price is due to it being an ‘assembly’, with clamping bolts and spacers assembled on the one-piece cable. Those parts are usually reusable, if you can have someone just make up a new cable for you.