I have been able to show that $$\log n! = \int_{1 + \epsilon}^n \log x \, d\lfloor x \rfloor$$
but I have not been successful trying to transform this Riemann-Stieltjes integral to an ordinary Riemann integral. The goal is to get rid of $d\lfloor x \rfloor$ via integration by parts, which should produce the terms $\frac{1}{2}\log n$ and $\int_1^n \left(x - \lfloor x \rfloor - \frac{1}{2}\right) \frac{dx}{x}$.
How exactly do these terms arise? I keep screwing up my computations. I've tried expressing this in a couple of different ways, but I don't seem to get the answer I'm looking for.