the excited electron by the cathodic ray will emit an X-ray in its return to the ground state
I expect 6 lines in the spectra (6 transitions to n=2 in the visible zone)
Why do you expect them to be in the visible light? You should calculate energies for all possible transitions, and count those in the visible range only.
But, broadly speaking, this is much more complicated. There are many reasons why some lines can't be detected - in general not all transitions are equally probable (so some are more/less intense than others), not all energy levels are equally occupied (which again leads to differing intensity, with some so dark they lie below detection limits), lines are never infinitesimally narrow, they will overlap, making their separation impossible - and so on.