Improvements

Because of the damage suffered by the secondary and the coupling issues due to the helical coil, I decided to  wind a new secondary and switch to a flat spiral primary, solving both problems.

Instead of painstakingly winding it by hand, I used a lathe with its autofeed. The spool of 28 gauge magnet wire unfortunately ran out before I was able to wind a full foot of secondary, so I ended up with a slightly shorter secondary coil. Difference in height is negligible though, tuning the coil should sort out any problems.

The toroid that I used is a scooter wheel wrapped in aluminum tape. This wheel is essentially the same wheel that was used for my previous toroid, only that unnecessarily bulky parts were taken off of this one. This particular toroid is a bit small for the size of the secondary, but considering that I searched for hours just to find something of the appropriate size, I think this one will do just fine.

Using a function generator and an oscilloscope, I was able to measure the resonant frequency to be 560kHz. This is higher than the previous coil, considering the fact that there are less loops in the secondary on this version. Resonant frequency is calculated by 1/2*pi*sqrt(LC), and if L goes up, resonant frequency goes up.

I made 4 small sections of PVC pipe to use as a holder for the primary coil. The primary wire has 8 turns maximum, and when finished, will be tappable each half turn.

The end product of all that winding looks like this. It's not circular, but Ampere's law should still hold.