The answer is 817.
At first I was just guessing, or rather letting Cinderella do the guessing for me:

But the magic behind this guess
that attempts to turn a floatingpoint number back into something like a rational is somewhat obscure (although the continued fraction in there are a nice subject). The idea is clear: all possible locations are the green area inset by 1 at every boundary. So that's $34\times 13$. The area where the circle does not touch is made up by two symmetric copies of the yellow triangle, which was measured in this image but which should be computed manually.
When doing manual computation, you can use the fact that all input lengths including the diagonal are rational (and in fact integers), so all trigonometric functions computed from these will be rational as well. You can compute the result stated above as
$ \frac{ \left(36 - 1 - \frac{36}{15} - \frac{39}{15}\right) \cdot \left(15 - 1 - \frac{15}{36} - \frac{39}{36}\right) }{(36 - 2) \cdot (15 - 2)} = \frac{375}{442} $
The parenthesized expressions in the numerator correspond to the legs of the yellow triangle. The first term is the corresponding rectangle side. The second corresponds to the one unit inset at the bottom left corner. The third removes the part on the wrong side of the diagonal. And the fourth represents the distance between the diagonal and the corner of the triangle, corresponding to the line one unit apart from the diagonal.
To understand the third and fourth term a little bit better, let us zoom in on the bottom right corner.

There you see two congruent triangles, both similar to one half of your input rectangle. Both are scaled such that one leg has a length matching the radius of your disk.
The whole concept of comparing areas assumes that the discs are placed uniformly over all possible positions, which strictly speaking isn't stated in the question.