Equilateral \triangle A B C has side length \sqrt{111}. There are four distinct triangles A D_{1} E_{1}, A D_{1} E_{2}, A D_{2} E_{3}, and A D_{2} E_{4}, each congruent to \triangle A B C, with B D_{1}=B D_{2}=\sqrt{11}. Find \sum_{k=1}^{4}\left(C E_{k}\right)^{2}.