Now that you've posted the schematic, we can see that you are trying to build a boost converter:
Unfortunately, this is a particularly crappy one. Run away. Find something better. Don't waste time trying to make this turkey hop along.
The basic problem with this circuit is that it relies on the non-ideal and unspecified characteristics of the transistors to work.
It's tough to follow the exact thought (if we can even call it that) process of whoever designed this mess, but here is how this circuit seems to work:
Start with everything off and quiescent. The switch is then closed. Q1 is turned on due to base current provided by the resistor. That turns on Q2, which pulls down on the bottom end of the inductor. That inductor voltage goes low. That turns on Q1 even more solidly thru C1 (Argh, add component designators to all components).
For some transistors, Q1 would stay on indefinitely, which keeps Q2 on indefinitely, which keeps voltage across the inductor. That can end several different ways, including burning out the inductor, blowing up Q2, or even blowing up Q1 since there is nothing limiting the current thru E-C of Q1 and then B-E of Q2.
What this circuit apparently relies on is the right combination of gain of the transistors, maximum current handling capability of Q2, the saturation current of the inductor, and the DC resistance of the inductor. If all these happen to work out, then the inductor saturates, Q2 can't keep up with the current required to keep voltage across the inductor, and the voltage on the C of Q2 goes up. This turns off Q1 thru the capacitor, hopefully before Q2 gets too hot and fries.
That switches off Q2, and the rest of the circuit is just a regular boost converter. The current thru the inductor can't stop immediately, so for a short time it squirts thru D1. That charges the output cap, which eventually builds up to enough voltage to run the LED, at least somewhat.
When the inductor current dies down, the C voltage of Q2 drops, which turns on Q1 thru the C1, and the process repeats.
Hopefully you can see how there are many things that can go wrong, and how this circuit relies on parameters not guaranteed in the datasheets of the transistors.
Trying to tweak this circuit for more LED current requires trial and error, and will likely leave a trail of burnt parts. Go find a real boost converter circuit that wasn't designed in the fit of a hallucinagenic stupor.