\(\text{Transformed graph has equation} \ y=\dfrac{x^4}{k^4}\)

\(\dfrac{d y}{d x}=\dfrac{4 x^3}{k^4}\)

\(\text{At }\ x=k:\)

\(m_{\text{tang}} = \dfrac{4 k^3}{k^4}=\dfrac{4}{k}\)

\(m_{\text{norm}} =-\dfrac{k}{4}\ (m_1m_2=-1)\)
 

\(\text{Equation of normal:}\)

\(y-1=-\dfrac{k}{4}(x-k)\ \ \Rightarrow\ \ y=-\dfrac{k}{4}x+1+\dfrac{k^2}{4}\)

\(\text{When }\ x=0, \ y=1+\dfrac{k^2}{4}\)

\(\text{\(y\)-intercept at }\left(0,1+\dfrac{k^2}{4}\right)\)

\(1+\dfrac{k^2}{4}=5\ \ \Rightarrow\ \ k^2=16\)

\(\therefore k=4\ \ (k>0)\).