Using Circuit of Plasma Globe to Power a Tesla Coil
Tesla coils work off of resonance between the primary and secondary circuits.  The resonant frequency of the primary tank circuit is set by the primary inductor and capacitor and is designed to match the resonant frequency of the secondary.  In your case, you have an existing driver circuit which may or may not match the resonant frequency of your coil.  If you've tuned a coil before, you know what the effect even a small mismatch in frequency can have on the output, so I doubt you'll see much if you hook it up.
Edit: also, plasma globes use maybe a few watts and even a small Tesla coil will take a few hundred, so your 21.5cm secondary setup will be several orders of magnitude underpowered.