I have solved the longer diagonal to be 2 if you plug in 1 as the side length of the hexagon, but I do not know how to calculate the shorter diagonal's length
If you draw the 6 "long diagonals" you divide the hexagon into 6 equilateral triangles. That is why the length of the "long diagonal" is 2- it is made of two of those sides of length 1. The "short diagonal" is made of two altitudes of equilateral triangles. You can find the length of an altitude of an equilateral triangle by observing that it divides the equilateral triangle into two right triangles have a side of length 1 as hypotenuse and half a side, of length 1/2, as a leg. Use the Pythagorean theorem to find the length of the other leg, the altitude.