The tube was working originally, but then the cement failed and took the wires going into the tube with it. 

Also, I think my electronic ballast is low-frequency, because I was able to use a 60Hz Stroboscope disc to check the speed of my stereo turntable under it, but I don't have an oscilloscope to find out.

The fact the light has a 100/120Hz flicker does not mean the ballast uses low ftrequency. It just means the HF ballast output is ampklitude modulated by the rectified mains voltage.

The only point in using electronic ballast is to utilize the efficiency, size and mass advantage of high frequency magnetic components. The point is, if at 60Hz a choke for certain power lamp, so given current and voltage drop weights 0.5kg, is 10cm large and dissipates about 5W, the same voltage and current choke designed for 30kHz is 15mm small, weights few grams and has barely 0.3W losses. Of course, the second needs the electronic converting the 50/60Hz feed into the 30kHz power for feeding it, which brings some extra losses, size and mass (in fact the HF generator is what forms the most part of the electronic ballast), but still you will end up maybe with similar size, but with less than 1W losses and barely 100g total weight.
So making the complex electronic operate at low frequency will kind of defeat the main point of their bare existence, when the magnetic components would have to be large anyway...