Solution:

Let $k = \sqrt{120 - \sqrt{x}}.$ Since $\sqrt{x} \ge 0,$ we need $0 \le k \le \sqrt{120},$ so $k \in \{0, 1, \ldots, 10\},$ giving $11$ values.

Each $k$ yields $\sqrt{x} = 120 - k^2,$ and since $120 - k^2$ is positive and strictly decreasing, the resulting values $x = (120 - k^2)^2$ are distinct.

Thus, the correct answer is E.