Just how does modern surveying equipment work? One guy holds one of those
mirrors (actually a prism) that reflects light back to the source,
independent of the angle of the mirror. The instrument head contains a
laser. Distances must be measured by angles, but how?

Akshully, it is not a prism -- it is more like a corner relfector whereby
any incident energy entereing the front is reflected back toward the source.
In case of the ranging instrument using a laser. The transmitter sends out a
discrete pulse of energy and measures the time it takes for the pulse to
return to a colocated receiver. The round-trip time is divided in half then
multiplied by a constant approximating one nano-second per foot to reveal
the slant-range distance. Angles ain't needed unless the surveyor must
calculate the horizontal distance, in which case he needs the alpha-angle
and that is done with a protractor.