As x gets large k quickly moves towards a value of around 0.658 something. Is it possible to say what value k takes as x tends to infinity? Is it a special number, like pi/4 or what have you?

Yes, that’s precisely the limit that answers the OP. The limit I gave was for the ratio x! e^(x) / x^(x + 1/2), which follows directly from Stirling’s formula without additional manipulation of limits with logarithms and L’Hospital’s rule. When I went back and applied those techniques to the ratio in the OP, the limit quoted by Cabbage fell out immediately.