For $k\in \mathrm{ℝ}$, let the solutions of the equation $\cos \left({\sin }^{-1}\left(x\cot \left({\tan }^{-1}\left(\cos \left({\sin }^{-1}x\right)\right)\right)\right)\right)=k,0<\left|x\right|<\frac{1}{\sqrt{2}}$ be $\alpha$ and $\beta$, where the inverse trigonometric functions take only principal values. If the solutions of the equation $x^2-bx-5=0$ are $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$ and $\frac{\alpha }{\beta }$, then $\frac{b}{k^2}$ is equal to ______.