I agree with the answer 1/(2√29) (=√(29)/58).
To see this, you need to know that the distance d from a point (m,n) to a line ax+by+c=0 is given by the formula .
Look at the line 2x-5y=0. This passes through the lattice point (5k,2k) for any integer k. Apart from these points, the lattice point that come closest to this line are those like (2,1) and (3,1) (as you can easily see by drawing a diagram). Their distance from the line is given by the above formula as .
Now think what happens when this line is shifted to a parallel line. So long as the distance shifted (meaning the perpendicular distance from the original line) is less than , the shifted line will still be within that distance of the origin. But as soon as the distance shifted is greater than that, the shifted line will be within that distance of (2,1) (if it is being shifted upwards) or (3,1) (if it is being shifted downwards).
The same reasoning applies to any other line with gradient 2/5. It will always be within distance 1/(2√29) of (infinitely many) lattice points.