Atari joysticks are pretty simple. All they are is a joy with four actuators on the bottom that press thin metal plates onto contact points. I believe that there is a thin layer of plastic over the contact plates as well. I haven't had one apart in years and years, but I remember opening one up and playing a game "intellivision style" when I was young.
Unless the plates are cracked or broken, you may be able to "fluff" a bent plate which will make it work again. If one of the actuators is broken, replace the stick from another joy and woila, you have a functional stick.
I replaced my 2600 joys with Wico sticks and used them on my C64. They had over ten years of active service, and they still work like new. I always liked those 2600 sticks, though. Maybe I'll pick one up and dink with it for a while.
APf