The original corded Xbox controller usually breaks right in front of the controller where the cord gets bent the most. I always leave about three inches of cord sticking straight out, then loosely coil the rest around the controller for storage. It makes the cords last a lot longer.
You can fix them by opening up your controller and cutting the cord shorter, bypassing the broken area then re-soldering the wires inside the controller. (The cords are basically USB cords. Four or five wires with a ground/shield depending on the brand.) I've done it several times. It takes time, but it works.
Because I used to host Halo LANs in my home and I have around twenty controllers and they have been well used. Only a few have broken in the last nine years, thanks mostly to storing them like I described above. If you decide not to fix your old ones and are planing on throwing them away, let me know.