How did Enrico Fermi calculate the classical Fermi Problem?

Well, I'm getting an answer about an order of magnitude too large so I must be doing something different, but here's my guess:

Blast wave travels at about speed of sound - 40 seconds -> 14 km in radius at this time. The paper is moved 2.5 meters by the wave - so the effect of the bomb is to displace a hemispherical shell of air of volume $2.5\textrm{ m}×2\pi×(14\textrm{ km})^2$ Multiply by 1 Atm to get energy of $3×10^{14}\textrm{ J}\simeq 80 \textrm{ kt TNT} $

I describe this in my book "Guesstimation 2.0" (Princeton University Press, 2012). The work done by the expanding shock wave is pressure times change in volume. The change in volume is as described by the previous answer: $2.5\,\mathrm{m} \times 2\pi \times (16\,\mathrm{km})^2$.

Fermi could feel the extra pressure due to the explosion, we can only estimate it. The extra force on his body must have been more than 100 N (20 lb) and less than $10^4\,\mathrm{N}$ (2000 lb) so we will take the geometric mean and estimate $10^3\,\mathrm{N}$. The extra pressure then is just $10^3\,\mathrm{N}$ divided by the typical frontal surface area of a person of $1\,\mathrm{m}^2$ or $P = 10^3\,\mathrm{N}/\mathrm{m}^2$.

Then $E = P \Delta V = (10^3\,\mathrm{N}/\mathrm{m}^2)(4 \times 10^9\,\mathrm{m}^3) = 4 \times 10^{12}\,\mathrm{J} = 1\,\mathrm{kT}$

Now multiply by a few because the energy of the bomb can go into light (photons), nuclear radiation, shock wave, ground pulse … and we get an estimate of 4 kT.

Fermi was there and he estimated 10 kT so this is in the correct ballpark.