Lakie is given two side lengths and the measure of the angle between them. How many unique triangles can she form given this information?

She can form only one such triangle. The given angle 'fixes' the shape of the triangle. Once the angle is given, the third side is also fixed.

So we have the first 2 sides, say a and b, and the third side c, fixed by fixing the angle between a and b. By Side Side Side congruence postulate, any triangle with sides a, b, c will be the exact same triangle as the one we constructed.