Singularity of $B$-field in a Dirac String

I was assigned this question related to Dirac strings:

Given a vector potential $vec{A}= frac{1-cos(theta)}{rsin(theta)}hat{phi}$, show that there is a singularity in the $B$ field proportional to $Theta(-z)delta(x)delta(y)$ on the $z$ axis. ($Theta(x-x_0)$ is the step function with its jump at $x_0$) Find the proportionality constant.

My attempt at a solution:

So showing there is a singularity simply results from the fact that at $theta=0$, the vector potential explodes because the denominator goes to zero. The same holds true for $r$. My question becomes quantifying the magnitude of this singularity via this proportionality constant. I’m assuming it’s essentially a measure of how quickly the field increases close the the $z$ axis, but I’m not sure how to procede from here. I’ve looked at quite a bit of literature on the matter, including Dirac’s original paper, but they all simply state there is a singularity, and make no statement about the size of it. Any insight that can be offered about the nature of this singularity would be deeply appreciated!