Solution:
If the sides are $x$ and $y$, we have $2x + 2y = 10$, so $x + y = 5$, and $\sqrt{x^{2} + y^{2}} = \sqrt{15}$, so $x^{2} + y^{2} = 15$.

Squaring the first equation gives $x^{2} + 2xy + y^{2} = 25$, and subtracting the second equation gives $2xy = 10$, so the area is $xy = 5$.