Why not just use a neon-leak tester? It is a oudwin/tesla coil capable of delivering up to 50Kv.
A place to find them is at a neon sign store/shop. You are looking at $80-$150 new, or you can build one easily using a car ignition transformer and a transistor.
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Seoul, This not ment to be a criticism of your idea, just a explanation of the physics of why we use the Oudin for boost, instead.
The Kettering (He Founded what became AC/DELCO) coil is a step up autotransformer, and does not run at a high Q (Quality factor of resonance.) Also its HV secondary is connected to its primary down to DC, and that is bad, leading to the BIG REASON:
FOR SAFETY:
The Oudin probe fits into the laser with a isolated piece of metal. Thanks to the "Ebonite" insulator in the probe. So you can have your probe near the anode of a device that when it lights is drawing up to 1700 watts. The ION laser tube is directly coupled to the AC line, via the rectifier and regulator, thus the anode is usually 90V -106V dc and both sides feed back to the AC line one way or another.
The Oudin is man rated, ie a neon tester just needs to make a weak RF field for testing a gas flask or a pumped neon tube that is NOT connected to a power line. For starting the laser, you want the known safety of the designed for the task Oudin!
Some other reasons:
A NLT does not return its field to ground through a internal capacitor to the line cord like a Oudin does. It returns through your hand, making your body the other half of a dipole antenna. Nor does the NLT have the peak energy to strike a cathode spot in a Argon.. The Oudin stores its energy in a fairly massive iron core, and dumps it into a ringing wave with a decay. The core is saturated when it does this, as when the core saturates, it pulls on a contact opening the charging circuit from the AC line ( its a mechanical AC buzzer), so you get a consistent sharp leading pulse.
Oudins have this "ebonite" HV plastic stick as a current limiter in the electrode structure.
The Oudin tends to handle huge capacitive loads better.
The Oudin is set up not to puncture the glass on a brewster stem at low settings.
(although we keep them way away from the junction between the silica window and the pyrex stem)
The Oudin sources just the right frequency and current to do the job of forming a cathode spot. Its running at say 300-500 Khz, much higher then a NLT, and way higher then a Kettering coil. Thus the chokes and cathode transformer keep it out of the PSU.
The laser's own ignitor, which is firing while this is going on, will pass through the Oudin's safey cap to the power line and you would not feel it at all. On a little NLT, all bets are off.
A Neon tester is OK for bare hene tubes. A ignition coil ran properly can glow a bare hene too. A ignition coil with a rectifier and filter cap and ballast resistor will probably lase a 1-2 mW hene (I know it will, done that years ago) A Oudin will light a hene via a small arc to the anode side of a Alden connector, usually lasing the tubes up to 20 mW or more.
If a Neon tester is held near the long brewster stem of a external mirror ion tube, you will see a very faint blue glow in the dark, If its held up to a sealed mirror metal-ceramic ion laser tube, the metal in the tube shunts the weak field and you get no glow. The oudin, used properly, will let you see a Argon or Krypton glow in room light in both tubes, by simply holding your other hand to shield the light a bit.
The gas pressure in a argon or krypton is usually much less then 800 millitorr, the neon tube is at 10 torr or more, and thus has a larger capture cross section, allowing the weaker NLT to couple to the gas better.
Note, on a tube not in a head, hold the side of the Oudin electrode up against the tube or brewster before turning it on, to prevent a direct arc from puncturing something. When done, turn it off before removing contact.
Questions?
Steve
