Answer: 21

Solution:

The base has area $36,$ so $\frac{1}{3} \cdot 36 \cdot h = 54$ gives height $h = \frac{9}{2}.$ By symmetry the sphere's center lies on the pyramid's axis, say at height $z$ above the base. Each base vertex is at distance $3\sqrt{2}$ from the axis, so the center's distance to a base vertex is $\sqrt{z^2 + 18},$ while its distance to the apex is $\frac{9}{2} - z.$

Setting $\left(\frac{9}{2} - z\right)^2 = z^2 + 18$ gives $\frac{81}{4} - 9z = 18,$ so $z = \frac{1}{4}.$ The radius is $\frac{9}{2} - \frac{1}{4} = \frac{17}{4},$ and $m + n = 17 + 4 = 21.$