A high-efficiency LED is probably your best bet in such a simple circuit.
But you should drive your mini-mosfet with a square pulse instead of a sine wave. Vary the duty cycle too, for best results. For a good strobe effect (freezing motion, etc) the "on" duty cycle should be very short.
I've made effective LED strobes using a 555 (either one-shot and externally triggered to synch to rotating machinery, or as built-in oscillator that can be set to arbitrary frequency). A 555 can be made to produce nice short pulses so that the strobe is effective in freezing motion.
For example see the circuit below. The basic timing signal (from signal generator, oscillator, or optical trigger, etc.) comes in to Pin 2, and the circuit drives an ordinary high-efficiency 5mm LED with enough brightness and short pulse width to work as a small stroboscope.