I built a Tesla coil in 1991
before the internet using this book.
My capacitor was two glass shower doors aluminum foil then some glass from storm windows for the top layer of glass. It glowed blue on the edges.
Spark gap was a music CD screws set on edges on circle saw motor with variac for speed control. Also used regular spark gaps I fashioned from things.
The coil that I got best results was a concrete form tube shellacked , I forget the size but lawn mower wheels fit in the ends and I mounted it to spin so I could wind on the 800 feet of magnet wire I got.
The arks when I ran the thing would skip winding so I unwound the whole thing and rewound it with fishing mono filament between the winding.
My PVC coil, with the heavy rubber covered wire worked well too, or more consistently. 10 gauge solid core of course.
Power was oil burner ignition transformers 23ma 10,000v alone 2 or 3 in parallel see what happens.
I got a brass ball from those things used in a fireplace
I think at one point I started running 220 through the ignition transformers and it worked better. Try 220 I know I did.
This was the early 90s and the good ignition transformers were getting phased out, replaced with these light cheesy ones. I guess today you just type "pole pig" into google and click shopping. Wanted one of those !
Anyway looking at your photo your magnet wire secondary may need a spacer between windings as I said it likes to skip winding when its close arks up the side. Fishing line then the shellac the whole thing.
And power the crap out of it, power in = power out. Play with a rotary and adjustable gap that's the variable for me that made up for math mistakes.
And make BIG capacitor, made one from that pile of storm windows I found that kicked butt till the electric started finding its was though and the foil was to close to the edges and would crack the panes.
Its so fun, my coil was nothing, got 10 inch sparks would be a YouTube embarrassment today but a running Tesla coil in real life is so much more impressive then watching on YouTube. Vent that ozone.
Have fun, don't get zapped that part sucks.