Why is the hydrogen the only atom that depends only on principal quantum number n and for all other elements, energies depend on both orbital quantum number and principal quantum number?
Incomplete. The reason is an "extra" conserved quantity, called the Runge-Lenz vector (check wikipedia). It's somehow, the same reason for which orbits close (i.e.: ellipses) under the 1/r^2 potential but they do not in general 1/r^p ones.
Also, if you take into account fine structure effects, there appears an "l" dependence of the energy levels. E.g., if you consider relativistic effects, as in Dirac eq.
The hydrogen atom has both spin-orbit coupling (coupling of the electron magnetic moment and the orbital magnetic moment) and coupling to the proton magnetic moment that create both fine structure and hyperfine structure of the electron energy levels. The famous ones are the Lamb shift and the 1420 MHz (21 cm electron-proton spin coupling) line in intergalactic hydrogen.
l dependence of hydrogen will also arise due to perturbations in the potential, such as an external electric or magnetic field. You can look up the Zeeman effect and the Stark effect.
SO! Really, only ideal hydrogen atom has only n dependence. Real hydrogen atoms have l and m dependence.