Congrats -- well done!
Generally speaking, if the tube lights to more-or-less normal brightness, but then goes dark, suspect the inverter. It's common for the insulation on the transformer's windings to break down. That causes excessive drive current, and a subsequent shutdown as protective circuits engage. These transformers are designed with very little margin (to keep them small), and it's a miracle they work as long as they do.
A typical peak output voltage in steady state might be a good piece of a kilovolt (several hundred volts rms; roughly proportional to CCFL length). To start up, the striking voltage might be 1.5x as high. So there's a fair bit of stress on the insulation of those tiny windings.
Generally speaking, if the tube lights to more-or-less normal brightness, but then goes dark, suspect the inverter. It's common for the insulation on the transformer's windings to break down. That causes excessive drive current, and a subsequent shutdown as protective circuits engage. These transformers are designed with very little margin (to keep them small), and it's a miracle they work as long as they do.
A typical peak output voltage in steady state might be a good piece of a kilovolt (several hundred volts rms; roughly proportional to CCFL length). To start up, the striking voltage might be 1.5x as high. So there's a fair bit of stress on the insulation of those tiny windings.