What would be a situation in which the lengths of two sides of a right triangle are known, but the measure of an angle is needed.

You didn't clarify what the angle is needed for, but still, I'd say never.

If you know two sides of a right triangle you can always find the third using the pythagorean theorem.

Once all sides are known, you can find the three angles: one is surely right, since you have a right triangle. The other two can be found using the sine law.