We have examined the conditions under which the breakdown of the Stokes-Einstein (SE) relation occurs in pure Lennard-Jones (LJ) fluids over a wide range of temperatures and packing fractions beyond the critical point. To this end, the temperature and packing-fraction dependence of the self-diffusion coefficient, D, and the shear viscosity, η_{sv}, were evaluated for Xe using molecular dynamics calculations with the Green-Kubo formula. The results showed good agreement with the experimental values. The breakdown was determined in light of the SE equation which we have recently derived for pure LJ liquids: Dη_{sv}=(k_{B}T/2π)(N/V)^{1/3}, where k_{B} is the Boltzmann constant, T is the temperature, and N is the particle number included in the system volume V. We have found that the breakdown occurs in the lower range of the packing fraction, η<0.2, and derived the SE relation in its broken form as Dη_{sv}=0.007(1-η)^{-5}η^{-4/3}(k_{B}T/ε)^{n}k_{B}T(N/V)^{1/3}, where n increases from 0 up to 1 with the decreasing η. The equation clearly shows that the breakdown mainly occurs because the packing-fraction dependence does not cancel out between D and η_{sv} in this region, which is attributed to the gaseous behavior in the packing-fraction dependence of the shear viscosity under a constant number density. In addition, the gaseous behavior in the temperature dependence of the shear viscosity also partially causes the breakdown.