Step 1: What you need:
2 x White or blue LEDs (25mA or better)
1 x Silicon diode (choose a schottky like the 1N5819 for better performance)
1 x BC327-25 or MPS751 PNP transistor (Do not substitute)
1 x BC337-25, MPS651, 2N4401 or PN2222/MPS2222 NPN Transistor (Do not substitute)
1 x 220uH coil inductor (see text)
1 x 10uF 10-volt (or better) capacitor.
1 x 1000pF (1nF, 102) capacitor.
1 x 10k resistor
1 x 1M resistor
9-volt battery and mounting hardware.
Most of these parts are available at AllElectronics Surplus and on eBay, here and here
If you want to experiment, you can try and make your own coil by starting with 15 turns on a toroid. If the circuit stays on by itself, add more turns; if the LEDs are dim, REMOVE turns.
Step 2: Assembly
The first picture below shows how I arranged the components. The next image shows the wiring on the underside. The third image is the 2 superimposed together.