Well, I simulated that original circuit in LTSpice and it doesn't even oscillate! I shorted out the LED, to make it pretty close to a standard two-transistor relaxation oscillator, and now it oscillates.
If you really want to stick with that circuit, you can add a buffer to drive the LED. You could also connect the LED (with a small-value current limiting resistor in series) across the 22 ohm resistor, instead of in the emitter path. But I think your main problem is that the current pulse is extremely short. This is a characteristic of this type of oscillator. In my simulation, it's about 2.3 ms which represents about a 0.1% duty cycle!
Increasing the ON-time of the LED will increase the apparent brightness. If the flash is too long, it won't look right - more like a burst of light than a short flash. But the point where it changes from a flash to a burst is much longer than 2.3 ms!
You can increase the flash duration by putting a resistor in series with the 10 uF capacitor. A value of 100 ohms increases the flash time from 2.3 ms to about 12 ms in my simulation.
I'm suggesting a number of changes to the circuit, so I've posted a full diagram with simulated waveforms. Here are a few comments on components and values.
The 10 nF capacitor isn't really needed. It's not present in the standard two-transistor relaxation oscillator circuit. Removing it has little effect on the circuit's operation.
The resistor I've added in series with the 10 uF capacitor determines the flash duration; higher value equals longer flash. It must be less than a few hundred ohms though. In the simulation, values higher than 680 ohms prevented the circuit from oscillating, so I would keep it no more than 220 ohms, for safety. For testing, you could use a 500 ohm trimpot, so you can play around with the flash duration.
The 33 ohm resistor in the Q2 collector circuit also can't be increased much, although it wastes power. The relaxation oscillator relies heavily on transistor saturation characteristics, and currents must be kept high, otherwise it simply won't oscillate.
The 18 ohm resistor in series with the LED sets the LED current, in conjunction with the power supply voltage. This value gives about 110 mA with the LED I used for the simulation, which is the closest I could find in LTSpice (it doesn't have a very good range of LEDs). You should choose the closest value that gives the amount of current you want with the LED that YOU are using.
I've put a 100 uF capacitor across the supply. I strongly recommend doing this, because of the high currents and fast transitions in the circuit. The circuit is quite fussy, and everything you can do to make things more comfortable for it will improve its chances of oscillating reliably.
