It all came down to trying to create an expression for the force that would act on the electron, due to the other charges in the atom.
Bohr's calculations work remarkably well for any one-electron particle -
In Bohr's calculations, it is necessary to come up with an expression for the force that acts on the electron due to the other particles in the atom.
With hydrogen, this was a simple matter, as the only other charge was the proton in the nucleus. The equation for this force is Coulomb's law, and is a simple expression.
However, for helium, there are two other particles (besides the electron we are trying to solve for). One is the nucleus (2 protons, 2 neutrons - no problem). However, the other particle that affects the behaviour of the electron we are trying to describe is the other electron!
Not only does this create the three-body problem, but in order to express the force acting on electron 1, we need to know the behaviour of electron 2, which is assumed to be identical to that of electron 1, and this is what we are trying to determine !
So, we need to know the answer to the question of how does the electron behave before we can even set up the problem of how it behaves!
The best that can be done in this situation is to use a method of successive approximations to estimate the electrons behaviour (and that is a very lengthy task indeed), and so, approximate results were all that could be obtained.
The amazing accuracy of Bohr's first calculation of the energy levels of hydrogen could not be reproduced as soon as one more electron was added to the particle.