Why excited states of Hydrogen atom are degenerate even with different angular momenta?

The Coulomb potential, along with conservation of energy and angular momentum, possesses another conservation law. Namely, the Runge-Lenz vector is conserved. Components of this vector are generators of a symmetry group. It is called a hidden symmetry or a super-symmetry. Because of this fact, clical orbits in the Coulomb field are closed, and energy levels of the Hydrogen atom are "accidentally" degenerate.