Solution:

Since $O$, $H$, $R$, $I$ and $D$ are distinct numbers from $\{1,2,3,4,5\}$, we have $O+H+R+I+D=15$ and $O-H-R+I+D=O+H+R+I+D-2(H+R)<15$. From this $O^{H^{R^{I^{D}}}}=\frac{(O+H+R+I+D)^2}{O-H-R+I+D}=\frac{225}{15-2(H+R)}$, hence $O^{H^{R^{I^{D}}}}>15$ and divides $225$, which is only possible for $O^{H^{R^{I^{D}}}}=25$ (must be a power of three or five). This implies that $O=5$, $H=2$ and $R=1$. It's easy to check that both $I=3$, $D=4$ and $I=4$, $D=3$ satisfy the stated equation.