Can broken cable be fixed?

It will take some doing! Follow this guide Apple USB SuperDrive Optical Drive Replacement once you have it opened up you should be able to de-solder the cable.

Now the fun begins! Using a 3D CAD program design a new strain relief block. Start by making an exact match to Apples part. Create a threaded end over the cable so you can thread on a threaded collar to hold the pieces together. Now split it in half (Top/Bottom) making room for the replacement cable you’ll want to use, and lastly create the collar piece. Find a friend or someone who can 3D print the parts (hopefully you have no errors) You may need to tweak the design a few times.