Troubleshooting my DIY Tesla coil setup

Question

Firstly let me say I'm relatively new to HV, so apologies for any silly mistakes. I made a Slayer Exciter circuit, powerful enough to drive a flyback transformer and make 3mm sparks. The circuit is powered by 12 volts at 0.5 amps, and it uses an unlabeled NPN power transistor in a TO-126 package (sorry, can't provide more details), a 10K resistor, and a 1N4007 diode, connected according to the circuit:

^{simulate this circuit – Schematic created using CircuitLab}

After some playing around with the arcs from the flyback transformer, I decided that I should connect the setup to a small Tesla coil. I wound a coil around two taped-together-and-secured-with-card felt-tip pens (outdated, see edit). Here is information about the coil:

Diameter: 1cm **outdated, see edit**

Length: 24cm **outdated, see edit**

Secondary: ~250 turns **outdated, see edit**

Primary: 1-10 turns (I tried various configurations) **outdated, see edit**

I managed to pick up a weak ~10MHz signal with my meter, but it cannot light up a neon "grain of wheat" light. Things I tried to troubleshoot:

1. Adjust resistor
2. Adjust number of primary windings
3. Reverse direction of primary coil
4. Reverse secondary coil connections (connect the other end)
5. Move primary closer to other end of coil
6. Reverse primary coil connections
7. Attach a top load
8. Simulate stray capacitance with small capacitor

I cannot get it to produce any visible arcs, or any effects for that matter, other than registering on my meter. Can anyone please help? What am I doing wrong (sorry for the bad photo)?

EDIT: Following the suggestion that was given in an answer, I rewound the coil for an aspect ratio of approximately 1:3. New coil specs:

Length: 9cm

Diameter: 3cm

Secondary Turns: ~100

Primary Turns: 3

Core material: corrugated card, rolled into a tube

Frequency: Undetectable

outdated picture, see edit

Answer 1

Tesla coils need to have a certain aspect ratio to operate properly. Due to its physical dimensions, the coil operates as a LC resonant circuit, using its surface area as one plate of a distributed capacitor. The other plate is provided by the surroundings (floor, ceiling, walls, your body, etc). If you make the resonator very long and skinny, its inductance goes up while its capacitance goes down due to the lower surface area. This makes the quality factor of the resonator too low to produce large-amplitude oscillations.

From my own experience, a Tesla coil should have an aspect ratio of between 1:3 and 1:5, although this is of course not definitive. You could, for example, build a coil with a diameter of 5cm and a length of 20cm.

1:24 is just way too skinny to work.