The tube shown in the picture of your set up is a CFL (compact fluorescent lamp) and may already have a starter and ballast inductor of some sort incorporated into it. If so, your circuit is not going to work. See
https://en.wikipedia.org/wiki/Compact_fluorescent_lamp
Re Duke37 comment. However, if your tube does not already have an incorporated starter, then in your set up your external starter completes the circuit to energise the heaters in the tube. But when the mercury vapour in the tube ionises, producing light, the heating elements now need to be disconnected. That is the 2nd job of the starter. So if the starter does not disconnect the heaters, the lamp will flicker out. You can overcome this by physically puling the starter out after it has tried to start. If the fault is the starter then the lamp will continue to produce light without the starter.
But of course as Duke points out you are fiddling around with lethal voltages and currents so you have to be careful. And looking at your set up, you are going to have your fingers almost touching the heater terminals and which will be at mains potential. It makes more sense to replace the starter with a new one when the setup is disconnected from the mains.