How to light an LED on 1 uS pulses
The LED is indeed blinking on and off with the pulse, just too quickly for your eyes to notice it.  The solution is to put a pulse extender circuit in place -- in other words, a monostable multivibrator.

Your garden variety 555 monostable will do the job with an inverter driving it, albeit barely -- you'll need to make sure the input to the 555 is driven hard LOW in order to get close-to-reliable performance out of it, as you are very close to the limits of the IC here.

A more reliable solution would be to use half of an 'HC122 or 'HC123 -- this gets rid of the inverter as you have an active-HIGH trigger available now, and is guaranteed to work worst-case by the datasheet specs of the IC.