Problems with shifter?
You have a Shimano Positron shifter and derailleur. If you didn't know you own a bit of bicycle gear shifting history. See Sheldon Browns Shimano page and look for 'Positron'.

Simply googling for 'shimano positron cable' should produce some hits. There appear to be some cables for sale on Ebay.

You should probably try to find out more about the Positron system, especially if there are different variants that might require different cables. I think there were at least a couple of models. 

Update 1: I did some google image searching and I believe you have a Positron II Six Speed.

Update 1.1: Further research reveals it's a Positron FH Six Speed. Positron II was 5 speed.

Update 2: Looking at how the cable is attached to the shaft, I think it is soldered. You may be able to re-solder the cable using a small butane torch. Or even use high strength epoxy to glue it back.