The neon bulb would have to be placed in front of the slit.
Typically you either have the neon lamp inside the spectrograph in front of the slit, or outside of the telescope so the light goes through the telescope optics and then the spectrograph. The advantage of having it shine down the telescope is you do not have to worry about the lamp position too much. If it is inside the spectrograph, then as I understand it the neon lamp must be carefully positioned so that the light follows the same path as the light coming from the telescope. Otherwise a shift in position of the lamp would cause an apparent but unreal shift in wavelength.
Though there were no light leaks, was the room completely dark or was there an opportunity for other light to pass into the spectrograph optics? Alternatively someone may have a spectrum of a similar lamp, in which case they may know if it also has a broad spectrum component.