Driving a LED high current / high speed (>= 200nsec pulse width, >=10us delay)
"Easy" electronically.
Does need to be designed.
Getting enough LED light into the target area at the required repetition rate "is the challenge". 

I have had to make many assumptions below - and possibly have made a few errors, or large errors, or humungous errors.  Regardless, its not important ! :-).
BECAUSE what I have done here is what you or someone else MUST do to design the system hat you want - and along the way you/they need to understand the points I've made and correct any errors, adjust any assumptions and see what the result is. 

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DRIVING the LED(s) will be easy enough.
Harder is getting the light needed into the area required in an evenly distributed manner. 

Many people have seen the astounding photos of bullets bursting through balloons eggs apples bananas and more - made by "Doc Edgerton" in the late 1950s and 1960s.
 What is not usually appreciated is HOW astounding they were are still are.

This classic bullet-through-apple photo, made in 1964, has an exposure time of about 0.3 uS (just about right for you) and/but was made with a high powered Xenon flash. 

Recycling such a flash in 5 us is 'challenging in the extreme'. [tm].
Replacing it with adequate PRF LEDs is "an ask".   

This photo and many similar from here



How to Make Applesauce at M.I.T. .30-caliber Bullet, 1964

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An LED is allowed a high current pulse AS LONG AS  this meets specs for abs max I and thermal issues. Modern phosphor LEDs often have a small ratio of Iabsmax/Imaxrated -maybe in the 1.5 to 3 range -  so the 25:1 MENTIONED BELOW MAY BE TOO HIGH. IR LEDs long ago allowed high peak/average current ratios and a data search will turn up LEDs that best meet your need. 

You need to say
what light levels you need,
how light is provided to target etc.
Essentially it may be as simple as "buy suitably large LED, buy suitable power supply, make simple suitable driver, do it".
Defining the parameters of IT are your job.
CROSS SECTIONAL AREA of light field (as in your image) is of crucial importance.
How light is detected is also important - what ligt level is adequate, how is the target drop imaged etc ?

Quick starter:

Assume:     

100 lux equivalent drop field illumination.
f/4 lens
20mm x 20mm area
Even illumination
0.2 us exposure time
100 lumen per watt LED efficiency.
100% LED light can be projected onto target area.

Power for steady illumination ~~~= 

1 lux = 1 lumen / m^2.  

Power lux x 1/(lumen per watt) x Area_m^2 x (aperture_f)^2
= 100 x 1/100 x (0.02 x 0.02) x 4^2 = 16/2500 ~= 6.4 mW LED input.
With units =
100l/m^2  x 1/(100 lm/w) x (0.02 x 0.02)m^2 x 4^2 (area compared to f/1) 

= 16/2500 ~= 6.4 mW LED input.    

That sounds rather good.
So far.

However, it assumes a sensor that can integrate the available light over a "suitable" time period. This may be 1/10s or 1/1000s or ... .
Shorter periods require more sensitive sensors for given light powers.
Or higher light power for given sensor sensitivity.  

A standard light level of 1 EV is defined as a 100 ISO sensor, 1 s exposure, 1 lux illumination, f/1 lens (Ginormous aperture). 

I've assumed a light level of 100 x that and a sensor of the same sensitivity 

To get the same amount of light onto the target field in 0.2 uS rather than 1 s(1,000,000 us) needs the light to be 1,000,000/0.2 = 5 million times as powerful.
5,000,000 x 0.0064 = 32 kW.  

At duty cycle of 5 us/0.2 us = 25:1 as shown that's a mean LED power of 32 kW/25 = 1280 watt mean.
That's some LED source.
It'S DOABLE but needs multiple LEDs, optics to focus light, very fancy cooling, and more. 

Reduce the area to 2 mm x 2 mm and get a power drop of 100 x
= 320 watts pulsed and about 15 watts mean.
Getting real.

2mm x 2mm "may well" be too small -> more power
100 lux may be more than needed -> less power
f/4 lens could have larger aperture -> less power - and depth of field around one nano gnats foot - which may not matter.    

Distance per LED-on  period:

Assume to start that drops are cubes of 25uM/side.

At 15 m/s max and 200 ns the drop will travel
Distance = V x t = 15 x 2E-7 m = 3 micrometer.
This is about 1/8th of a 25 um cubed drop. 
According to my E&amp;OE calculation that's about 15 picolitre so in your 3-180 pl range.  

I can see that it would be NICE if a drop near the small end of the range falling at Vmax would only be blurred by about 12% of its length by a 200 ns LED flash.
BUT it is probably far finer than you need.  

I suggest that you work out ALL the things you need and have (light intensity, target area, run time (eg a 1 or 5 s run and then cool down would greatly improve ease of cooling).  Then postulate basic light from LEDs and specs for source. Then discuss. 



Excellent Cree paper cited by Glen_geek
Diminishing returns set in quite rapidly.   

Pulsed Over‑Current Driving of Cree® XLamp® LEDs: Information and Cautions