The sqrt(2 pi) comes from a factor in the derivation of the Stirling formula that is similar to the error integral of the Gauss normal distribution, see Wikipedia (in German, but the formulas should speak for themself ;-) ). The trick Laplace or Poisson came up with, was to replace the integral by its square, ending up with an integral over a 2d domain which is parameterised in polar coordinates to perform the integrations.

The 1st step, squaring the integral, definitely is not really intuitive when doing derivations as a student.