I've never seen one fail like this.  The lamp lights up white when you put it next to globe which shows the vacuum is still there and the cathodes aren't blackened.  It just stays on all day long with the filament glowing.  Apparently the main lamp current source failed. 

http://www.youtube.com/watch?v=aQP6I71x57M

It is the short of the starting resonance capacitor (parallel to the tube), so the inverter current flow only trough the filament (that are normally in series with this capacitor).
It is quite easy to fix, you need ~4.7..6.8nF/1200V foil capacitor.

But the question is, if the capacitor breakdown was not a consequence of the tube running out of the emission mix, so the tube ignition voltage was too high for the capacitor to tolerate it. This would be supported by the fact, then during the filament glowing they do not enter the thermionic discharge mode (so ends glowing white in the lamp's color).