How to replace a Sturmey Archer's cable?
You are correct. This is different than "regular" cables. On regular cables you pick the end you want and cut off the other. Thats because the connection on the cut-off end is usually a clamp. You run the cable under a screw/nut and tighten it down. 

Then you just put a cable crimp on the end to protect the cut cable


However, your internal hub cable is differnet. The cable and housing is really one piece. You have to replace it all together. I have a similar issue on my Shimano internal 3 speed hub. Luckily you can get an entire replacement set (cable and housing).You can get something like this from Harris Cyclery (I only hype them because a Sheldon Brown article was helpful to this post.)



You'll have to make sure that the barrel is the same threading as your "adjusting part" and that you get the length you need. A sales person (or phone call) should get you the exact part. It would probably help to know the model of your hub. It should be engraved or stamped somewhere visible.