Assuming the paper shifted perfectly laterally, I assume you'd just be measuring the difference in the volume of half a sphere ten miles in radius, and a sphere ten miles and 2.5 meters in radius. I think you could use the ideal gas law at that point to figure out the energy.
I think you're right. Ideal gas law wouldn't apply to the plasma sphere but knowing shockwave strength at some distance you can figure out how much energy was transferred to the atmosphere.
I agree the calculation probably worked by measuring the kinectic energy of that half-shell by measuring its velocity*thickness directly using pieces of paper.
Perhaps the more important thing is that things are happening on very short time-scales, far enough from equilibrium and temperature might not even be well defined.
I'm just guessing though.