This is probably too late for your use, however, I thought I remembered this from somewhere. It is covered in chapter 1 in "Transport Phenomena" by Bird, Stewart and Lightfoot. The graph in the book indicates that epsilon is directly related to the energy. With the info you have, I would guess you would then have to take the first derivative of the LJ equation and set it equal to 0 to solve for the other constant. That would then give you both constants to solve the equation directly for each distance.