This gives an idea
In[18]:= Arg[-2 - I # \[Pi]] & /@ Range[10]
Out[18]= {-\[Pi] + ArcTan[\[Pi]/2], -\[Pi] + ArcTan[\[Pi]], -\[Pi] + ArcTan[(3 \[Pi])/2],
-\[Pi] + ArcTan[2 \[Pi]], -\[Pi] + ArcTan[(5 \[Pi])/2],-\[Pi] + ArcTan[3 \[Pi]],
-\[Pi] + ArcTan[(7 \[Pi])/2], -\[Pi] + ArcTan[4 \[Pi]],-\[Pi] + ArcTan[(9 \[Pi])/2],
-\[Pi] + ArcTan[5 \[Pi]]}
but here you see a rational positive k does it as well
In[24]:= FindInstance[Arg[-2 - I k \[Pi]] == -\[Pi] + ArcTan[k \[Pi]/2], {k}]
Out[24]= {{k -> 106/5}}
why not compute it directly, it's just plane trigonometry.