Knights are 24/13 times faster than the king

On an infinite chess board, how much faster can the knight reach a square when compared to the king, in average? More generally, for coprime $b>a\in {\mathbb{Z}}_{\ge 1}$ such that $a+b$ is odd, define the $(a,b)$-knight and the king as \begin{equation*} \begin{aligned} $\mathrm{N}$_{a,b} = {(a,b), (b,a), (-a,b), (-b,a), (-b,-a), (-a,-b), (a,-b), (b, -a)},\newline $\mathrm{K}$={(1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1), (1,-1)} \subseteq $\mathbb{Z}$^2, \end{aligned} \end{equation*} respectively. One way to formulate this question is by asking for the average ratio, for $\mathbf{p}\in {\mathbb{Z}}^2$ in a box, between $\min \{h\in {\mathbb{Z}}_{\ge 1}\mathrm{\mid }\mathbf{p}\in h\mathrm{N}\}$ and $\min \{h\in {\mathbb{Z}}_{\ge 1}\mathrm{\mid }\mathbf{p}\in h\mathrm{K}\}$, where $hA=\{{\mathbf{a}}_1+\cdots +{\mathbf{a}}_h\mathrm{\mid }{\mathbf{a}}_1,\dots ,{\mathbf{a}}_h\in A\}$ is the $h$-fold sumset of $A$. We show that this ratio equals $2(a+b)b^2\mathrm{/}(a^2+3b^2)$.