Haven't been able to work this lately because of available time.
JKT made some good points about the required illumination being ~400lm for proper exposure, so working with that as a reference.
400lm of continuous light using LEDs with 100~150lm/watt requires 3 to 4 watts. Assuming 4 LED modules this equates to only 1 watt per LED Module, heat sinking should be relatively easy at this level.
For flash use, a 1ms flash burst would require 400lm/T, where T is the flash period. This is 400,000lm, likely way too much for practical consideration. At 10ms then 40,000lm are required and this can be achieved with 4 or 8 LED modules of 10,000 ~ 5,000lm per LED module.
A typical speedlight has ~60Ws of available energy (1/2 CV^2). Xeon tubes are ~50% efficient, with another ~10% lost to I^2R heating in the capacitors and wiring & tube resistance. So a reasonable estimate of optical output is ~25Ws.
High power LED modules usually require 38~43V and with 100~150lm/watt would require 2~3 amps for the 10,000lm modules and 1~1.5amps for the 5,000lm models. This is well within the capability of the controller previously mentioned.
4 LED modules with 10,000lm per module would produce a total 40,000lm, and require under 2 watts average power per module to produce 100lm per module continuous or with flash. The peak optical power is limited to 10,000lm per module at ~100watts.
So an optical LED flash could produce ~40,000lm and ~400 watts peak power, whereas the speedlight would produce ~25Ws. The time cross over point where the LED modules equal the output of the speedlight is 25Ws/400W or 0.0625 seconds. At this time the LED module would dissipate an average power of 16watts. It would take about six times more LED output to equal a speedlight output at 10ms, or ~ 250,000lm, which would require ~25 10,000lm modules!!
This may be the reason there are no high power LED based speed lights or strobes available
As JKT mentioned earlier, may require too many LED modules to get a good LED flash output power in millisecond ranges, and the high power LED modules aren't cheap either
Need to think about this more, maybe others will comment.
Best,
Research is like a treasure hunt, you don't know where to look or what you'll find!
~Mike