Looks better than some of my projects.
Scavenging tv sets is a great way to acquire a bunch of parts.
Several sets of jumper clips (short and long) are one of the best things I ever got.
I guess that's the spark coil at the right.
I'm still wondering about the length of the waveform you are applying to it, and the frequency.
Old fashioned automotive ignition systems had 'points' inside the distributor. The points were opened and closed by a mechanical cam (6 or 8 sided). There was an optimum amount of time they were supposed to be closed, in order to conduct ample juice to the coil.
The purpose is to create a strong spark in the cylinder. So there was a meter called a dwell meter, to make sure the points were closed for a certain number of degrees of distributor rotation, per cylinder. Usually it was 40 to 45 degrees. The engine revolved twice per one revolution of the distributor cam.
The system is designed to deliver a high voltage spike at the secondary. Same as you're trying to get. But it is important to minimize heat buildup in the primary power loop. I once removed a spark plug wire to check the timing, then forgot to put it back on the plug. The spark could not be delivered to #1 cylinder. I drove the car and after five minutes the ignition module burned up. Stopped the car dead.
To make a Jacob's ladder, I believe the goal is to achieve a close duplicate of the automotive ignition system. Make sure the current is high enough, the dwell is long enough, the frequency is right, keep the spark gap small, etc.
There is also such a thing as 'souped up' ignition systems. It's supposed to provide a stronger spark. Might be worth a look at how they work too.