Answer
h=6
Work Step by Step
Solve for $(A-5I)x=0$.
$\begin{bmatrix}
0&-2&6&-1&0\\
0&-2&h&0&0\\
0&0&0&4&0\\
0&0&0&-4&0
\end{bmatrix}$~$\begin{bmatrix}
0&-2&6&-1&0\\
0&0&h-6&1&0\\
0&0&0&4&0\\
0&0&0&0&0
\end{bmatrix}$
The only way for the dimension of the eigenspace to be 2 is for this to have 2 free variables. This is only true when h=6
