Why is this integrand not integrating to a Bessel function?

I know from the identities of Bessel functions that the following is true: $ $ J_{m}\left( x \right) = \frac{ 1 }{ 2 \ \pi \ i^{m} } \int_{0}^{2 \pi} \ d\phi \ e^{i \left( x \cos{\phi} \ – \ m \ \phi \right)} \tag{0} $ $

However, when I try to integrate the following, Mathematica doesn’t actually integrate the result. It just spits back the input.

Assuming[Element[k, Reals] && Element[m, Integers], Integrate[Exp[I (m x + k Cos[x])],{x, 0, 2 Pi}]] 

Should I allow m to be more general, i.e., any numeric value (not necessarily fond of fractional Bessel function indices) or am I missing some other assumption that is preventing Mathematica from giving the proper result?

(I doubt this matters but for completeness, I have v12.1.1.0 and am using a Macbook Pro laptop.)

Note the linked question Evaluate the defining Integral of the Bessel functions of the first kind has several answers that kind of answer my question, but I am curious why the above does not actually integrate. That is, I specify the regions of validity clearly and arguments so Mathematica should recognize that the input is the integral form of the Bessel function. I want to know why it’s not evaluating, not just that it won’t or how to make it work by explicitly specifying the Bessel function index m using something like Table.