There are some tricks to sweeping the charge carriers out of the LED to speed up turnoff...
take a look at :
High-Speed LED Driver for ns-Pulse
Switching of High-Current LEDs
Hubert Halbritter, Claus Jäger, Rolf Weber, Michael Schwind, and Frank Möllmer
IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 26, NO. 18, SEPTEMBER 15, 2014
and do some Googling for LED Pulse PIV.. and you should be happy..
There are three schools of thought.. Avalanche Pulse, Bipolar Driver to sweep out charge by reversing polarity, and brute force high speed switching with a small series resistor. The above paper uses brute force, a fast mosfet driver, and adds an inductor which speeds up turn on and turn off... There is demand for this, with at least one research group developing a die with seven mosfets to do the current switching, charge sweeping, and shunting the led to correct for issues in turn on-turn off time.
If you want a nice sharp square wave with rise and fall times in nanoseconds, this becomes a LED chip and packaging modeling exercise, ..
This one is good, as well...
Pulsed operation of high-power light
emitting diodes for imaging flow
velocimetry
C. Willert1, B. Stasicki2, J. Klinner1 and S. Moessner1
2010 Meas. Sci. Technol. 21 075402
Avalanche Pulsing is easy, until you need to clean up the trailing edge of the pulse...
Both of the above papers use available parts, no special magic... Digikey sells some low ESR, medium voltage, moderate capacity, lytic caps for pulse duty, and one paper makes mention of them. Board layout and material choices are crucial, with low impedance, high current, low inductance leads needed, as well as a "stiff" low ESR power supply.
Steve