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Measurement of pulse amplitude - suggestion for ADC

I have a pulse train (0Volt - X Volt) of regular time but varying amplitude (to pulse a LED). I have to measure the amplitude X of the pulses.

Timing is shown in the figure below. Times may change but these are roughly the values. The pulse is flat topped and one sample at the end of the pulse is sufficient.

Resolution of 14-16bit is fine. The issue as I understand is that the adc need some kind of sample and hold and a syncronization. Trigger is available. Interface of I2C is prefered, but also SPI can work. Not parallel.

I invite suggestions on a suitable ADC.

br henning