How does this Miniature Solid State Tesla Coil work?
Your primary circuit serves as a high frequency oscillator to provide the excitation of the coil. 

But to get optimal response from your Tesla coil, or perhaps any response at all you need to match the resonance of the secondary circuit with the driving frequency of the oscillator. 

Number of turns (inductance) is only part of the resonant solution of the secondary circuit. You also need capacitance in the secondary. 'Coilers' (the name given people that build and play with Tesla coils) will usually use a 'top load' that serves as the capacitance. You can use a bead or a ball but more commonly a torus is used. 

Furthermore geometry of the air gap transformer (the two coils) is important. It's not enough to have just the number of turns on the primary or secondary. The design should provide good electromagnetic coupling. The length of the secondary should be about a quarter wavelength of your design Tesla coil frequency. It should also be a narrow diameter compared to the primary coil which is usually wound flat in a spiral. The primary usually also has a tap to allow tuning of the coil, and the primary placed at the base of the secondary to project an upward field onto the secondary. Other coilers have used co-axial cylinders or cones to provide the coupling.