Even if the original poster has solved this problem long ago, this sort of situation does come up frequently so here is my answer, over five years later...
This is similar to problem SLAC had when designing the "Z zero Factory" in the 1980s. Very narrow pulses would arrive from beam position sensors. Their amplitudes had to be digitized for use in steering the beams. No beam intersections, no Nobel Prize! I spend about a year working with the EEs and techs calibrating the electronics, so I've seen this before.
The pulse were first put through a passive low pass filter, specifically a Gaussian filter that would produce an approximately Gaussian-shaped (iirc) pulse with a peak occurring several nanoseconds after the input pulse. Pure LC circuitry, with no noise, slew rates or anything else to screw with the measurements. A sample/hold was triggered at the expected time of that peak. The delay of the Gaussian peak relative to the original pulse gave time for the trigger circuitry to work. The S/H used a diode bridge powered by, as I recall, four transistors arranged in an H-bridge, like a motor controller. The pulse edges of the S/H did not have to be anywhere as sharp as the edges of the original input pulse, though sharp by any ordinary standards.
We did not try to make diode-based peak detector as in the diagram you show. The forward bias of the diode would make calibration at lower beam intensities (smaller pulses) difficult. You overcome forward bias with an op amp, a standard textbook circuit perfect for such things in audio and sub-MHz applications, but for accurate measurement, forget it. Gain-bandwidth product, slew rate, stray capacitance, blah!
Good results are obtained with passive filter pre-processing, then using digitization techniques that are still quite fast but able to handle pulses on a scale of tens of nanoseconds.