the diagonals are perpendicular and bisect each other.
midpoint of the given diagonal is (-4,5), so the other diagonal goes through there.
slope of given diagonal is -8/4 = -2, so the slope of the other diagonal is 1/2.
the sides of a rhombus are all equal, so we need to find (h,k) such that it lies on the line through (-4,5) with slope = 1/2, and the distances from (h,k) to the two given endpoints are the same.
However, that doesn't help much. All points lying on the perpendicular bisector of the given diagonal are equidistant from the diagonal's endpoints!
A rhombus can be anything between a square and a squashed-flat line-like narrow thingy. All we know is that the sides are all the same length.
Without knowledge of the side length, area, or the angle between sides, or something, we are stuck here.