One of the many, old Sturmey Archer parts worth fixing is the "click" shifter, especially 1930s-60s era shifters. The quality of these shifters is quite good, and they often need only minor repairs to be fixed.
Over the weekend, I fixed three such shifters using the guts of a damaged shifter I had sitting in my parts bin.
It's not all that difficult. You drive out the binder pins running through the shifter from the back to the front with a small punch. Once apart, notice how all the parts line up. Clean up everything, removing old grease and rust. Usually the issue is a broken or weak flat spring. Either replace the spring with a spare or gently tighten the spring by pinching it together with your fingers, a clamp, or a vise. Lightly oil everything and reassemble.
The most important step in re-assembly is to put the binder pins back into place. You do this at first by hand: put the pin in and press it through to the back with your fingers. It should be very close to just going all the way through by hand. It may just, barely stop at the back plate. Make sure it is lined up with the holes in the back of the shifter.
Once that is done, you use your hammer and an anvil or similar metal plate to tap the pin faces from the front with a hammer and properly sized punch, so that the backs of the pins strike on your anvil or metal. This will flatten the backs of the pins back out and the shifter will be back together.
BikeForums.net user "BigChief" has a very good write up on this subject as well, here: http://www.bikeforums.net/19725462-post13206.html.