Overdriven LED's vs electronic flash : Feasibility
Firstly, is the Xenon flash long lived enough for you?
If it lasts 50,000 flashes and you take one photo per minutes then that's:
50,000 / 60 / 24 = 34 days, 24 hours a day. 

Whether an LED will work for you depends on how much light output you want.
If an area of a few square metres max is enough then  LED lighting may work.
Note that it is not usually possible to substantially over-drive modern power LEDs - more on that below. 

An eg 5 W LED will provide say around 600 Lumen. Closer to 1000 lumen for the very best.
But say use 500 lumen of usable light.

A typical modern LCD monitor provides 300 lux at the screen surface - that's 300 lumen per square metre.
If illumination equal to screen brightness is bright enough for you (as I'd expect it would be) then you can illuminate (500 available lumen) / (300 lumen/m^2) = 1.66 m^2. Say 1.5 m^2.

You can adjust those figures to suit what you need.

lux = lumens/Watt x Watts / Area m^2.

You need to collimate / focus / distribute the light to illuminate the area of your choice.
If the area is small and can be close to the LED then a single 5W white LED is liable to work.

Lifetime of good quality power white phosphor LEDs will be over say 20,000 hours or about 2.5 years continuous. If you use the LED only say (1/1000 s)/60s = 60,000:1 life extension it should last to the far end of the next ice age. (Over 100,000 years) all else being equal.

In practice the LED will be subject to thermal shock, other aging factors and the death of the experimenter. Odds are it should last a lifetime if run in spec. 

To put things in perspective. If you wanted 300 lux at 10 metres range with your LED illuminating a half sphere. Area = 4PiR^2 =~ 1200 m^2. At 300 lux that needs 360,000 lumen and at 150 l/W that would require 2000 Watts of LEDs if run in spec. Which shows why Xenon flashes still have their place, even at ranges like this 



100+ Watts of LED illumination at 1/1000s would probably be required for this photo IF the 'flash had a moderate degree of focus. 

If the scooter is at 5 metres and the flash covered a half sphere you'd need around 500 Watts of LED lighting for 1/1000 second. BUT only 50 Watt at 1/100s.

Not quite gratuitous image using flash illumination. 
Black square on young woman's cheek is 1% of image height - see text. 

Larger



Over driving LEDs.

It was traditionally possible to drive eg Infra-red LEDs in pulse more at currents and power output levels far in excess of their continuous ratings. This high overdrive capability is NOT commonly found in high power white phosphor LEDs. 

Modern high power white phosphor LEDs usually have absolute maximum allowable currents not much above their rated maximum operating currents. A ratio of less than 2:1 would be usual and much less than 2:1 is not unusual. 

I have driven a few samples of such LEDs to several times their absolute maximum ratings for short periods. While no obvious immediate damage occurred at ratios of say 3:1 to 4:1, it seems likely that such operation would substantially shorten LED lifetimes. 



The Elephant in the room:

The above 'calculation' assumed 1/1000s exposure time.
At 300 lux you'll need about ISO800 and f/1.4 to expose correctly 
In 1/1000s your rain drops will fall 1 mm per m/s of velocity.
Raindrops typically fall in the speed range 2-10 m/s - large drops fall faster.
So at 10 m/s a large drop will fall 10mm - which is very likely to be too far for sharpness.
If the image area is about 1300mm x 1000 mm tall then 10mm is 1% of the image height. 
In the photo above there is a black square on one young woman's cheek. Better seen in larger version. That is 1% of the image height - 14 pixels in about 1400 in the original. 

To get raindrops down to 1mm at 10 m/s in a 1 metre height frame will require an increase of light by a factor of 10. Still in the capability of LED lighting - but now more like 50 Watts at f/1.4 and 800 ISO.

I don't know what quality of image you need, but an eg Nikon D700 will take reasonable photos at ISo 6400 - 8 times as sensitive as in the above 'calculations'. That would allow you to use something less fancy than an f/1.4 lens and to get a batter depth of field - probably around f/4. Still not vast DOF - crank up the light :-).