$z-2i=1$ represents a circle with centre at $\left(0,2\right)$ and radius $1$ unit

$\left|2z+1\right|=2\left|z-\left(\frac{-1}{2}\right)\right|$ represents twice the distance from the point $\left(\frac{-1}{2},0\right)$

Hence from the diagram,

 $m=2AD=2\left(CD-AC\right)$
$m=2\left(\frac{\sqrt{17}}{2}-1\right)=\sqrt{17}-2$
$M=2BD=2\left(CD+BC\right)$
$$\begin{gathered} M=2\left(\frac{\sqrt{17}}{2}+1\right)=\sqrt{17}+2 \\ \Rightarrow {\left(M-m\right)}^2=4^2=16 \end{gathered}$$