cable too short to reattach into a sturmey archer 3 speed shifter
Cables have special ends on the brake lever / shifter lever side which you need in order to hook the cable into the lever. A brake cable end for mountain bikes looks like:



You can't add these ends to an existing cable. If the cable is broken near the lever, then almost surely the part of the cable with the end is not going to be long enough. Thus, if you don't have a cable end, buy a new cable.

It's quite dangerous to have a broken brake cable, so you should get that fixed asap. 

If you do have the appropriate cable with end, you just need to remove the cable tension (disconnect it), and then thread it through the part of the lever which can grab the cable as shown here.