Timeshell the LC circuit is supposed to be between one choke and the "capacitor" or the sum of the coils (mutual inductance) and the "capacitor"?

Good question.

Treat each side of the WFC like it's its own capacitor.

So, if you measure your WFC (or calculate; measuring it should give you a similar result) one way, let's say it gives you 25pF. When you reverse the meter leads, it should give you a different result, such as 30pF. So, as an example, to have a frequency of 5kHz on both sides, the side that measures 25pF would need a coil of 40.528H and the side of 30pF would need a coil of 33.774H.

This doesn't include the secondary which you will also need.

BUT, there is more to it than that. You need to select a wire gauge that will give you a suitable resistance for the length of wire you use to create the coil for impedance matching to maximize power transfer to the WFC. And do some research into the reactance of capacitors and coils.

You really should measure each tube pair of your water cell and make sure that they are very close to equal values as you can reasonably get them. Then add up their capacitances using the equation for capacitors in series which should be the total value of your WFC.