Well, if you're willing to take it apart / clean it / lube it / reassemble it, that's pretty much what can be done.

Before you do that, check the cable. Cables can do crazy stuff!

One thing to check: make sure your speedo cable didn't have some kind of tight bend or kink in it. Make sure it does a nice wide arc going smoothly into the speedo housing, and that the cable seats properly in that housing. It's also possible for a broken cable to continue turning in a kinda pathetic way after breaking, because the two broken cable pieces are sitting against one another and the one might still be able to turn the other, kinda, some of the time.

So, go ahead and substitute another cable. See that the speedo still has issues on a known-good cable, before you go overhauling the speedo.