Leds

This isn't random leds as you are actually selecting one. Secondly there is a lot wrong with this circuit. The power supply needs to be grounded to even work. Also it is a dc power supply, it needs to be ac in order to pass voltage and current through the transformer. So I'd set it around 6 volts ac. The transformer should just be dialed down to have 1 turn in the primary and 1 turn in the secondary. So it's a 1:1. Then dial the resistor down to about 100 ohms. Next, dial the capacitor down to a reasonable number, somewhere in the nano (n) or micro (u) range. I believe I set it to 100 or 10 nf. Not at 10 kf like you have it, it will never charge the way you have it. Or just delete it, since there is no diode in front of it to block the negative cycle of the ac, it will discharge and recharge as the ac cycles, serving no purpose. This should make this circuit work. Atleast it did for me. What would be easier with this circuit is just making say a 5 volt dc power supply, wiring it up to about a 100 ohm resistor, wiring that up to a single pole double throw switch, then wiring up the switch to two leds like you have here. It would do the same thing, just no ac to dc to worry about.