I was one of those who VTCd so here are some minimal steps for me.
- Remove the screenshot as per this guideline.
- As you point out, there is an identity involving $\delta$-functions and exponentials. What is this identity and explain why can't you use it. It's a pretty standard identity so find it.
- momentum operators involve derivative, so what is this identity and why can't you use it.
- Do you need to use commutation relations. Does this help you or not?
I suspect that these last two points are where the conceptual struggle is located.
Maybe the following can help to identify the conceptual hurdle. Can you compare with ordinary 1d momentum operators or a harmonic oscillator with given frequency? In other words, if you started with $a^\dagger a$ and wanted to covert that to $x$ and $p$, how would you go about doing this? What is the dispersion relation and how does this come into play in relating $k$ and $\omega$? Is it important?
If you can re-edit your question so as to make it broader, and include the above elements so you clearly identify where the (conceptual) problem lies, then maybe your question will be reopened.
Remember that writing goods questions is hard, and writing good homework-like questions is harder. There's a thread for discussion, and another thread containing good examples, so be inspired by this.