While trying to figure out why my new diode tester wasn't running, I found a decades old sketch of the circuit.  It was stuck in between some other drawings that I had stuck in my copy of the "Engineers Notebook."

It turns out that you don't need the LED on the 3909, but you do need a pull up on the JK inputs.

I updated my prototype and the schematic accordingly:

It is still a very simple circuit, very well suited to building on a perfboard.

If you build a copy, I recommend you use sockets for the ICs.  Back in the day, we used sockets because ICs were expensive and hard to get (ever drive 40 miles to get to the nearest Radio Shack?)  Sockets let you swap an IC from one gadget to another without trying to unsolder 14 pins without toasting the IC or burning the pads off the board. Now a days, the required ICs are obsolete and again hard to get.  Putting the ICs in sockets lets you pop them back out if you want to build another circuit with them.