How to find L value of a coil

You can analyze the input signal as a sum of sinusoids. Maybe analyze up to the 5th or 7th harmonic. But there is an easier way..

To actually get a square wave current into a pure inductance is impossible- the voltage would have to be infinitely high to get the current to change instantaneously at the edges.

If you have some idea of the inductance (better yet, a fairly good model of the solenoid in terms of series resistance, parallel capacitance and eddy current losses), the easiest approach is to do a SPICE simulation (LTSpice is free) with an ideal current source to see where the voltages go, and what happens if the voltages are limited to some reasonable value. That will quickly give you a feel for the problem (if your model is reasonably realistic).

Edit: Below I've simulated a +/-100mA ideal current source into an ideal 100uH inductor flipped at 15kHz. The only thing non-ideal is the rise and fall times of the edges, which I've set to 1usec each in order to limit the voltage. As you can see, the voltage required to give you a rise time of 1usec for current change of 200mA (from -100 to +100mA) is 20 volts.

You can see that the voltage required to get the current to change from -I to +I or back in time tr = tf as |V+| = |V-| = \$ \frac {2 I L}{tr}\$

This is the formula for a simple solenoid: -

Basically if you space the (same number of) turns out, the length dimension (l) increases and inductance reduces.

As for a transformer core, adding an air gap significantly reduces the inductance. You have to think about the first formula and imagine that the solenoid was wound on a piece of ferromagnetic material such as ferrite or steel laminations. The dominant material that affects the inductance is not the core material but the air gap from one pole end to the other - imagine it like putting a 1k resistor in series with a 1 ohm resistor - the 1k dominates and represents the air gap; the 1 ohm is the core material and is totally dominated by the air gap.

For a closed transformer core a different formula is used because the core has no gap (or a very small gap) and the core dominates. However, introducing a small gap does reduce the effective permeability of the core significantly.

Usually this is specified on core data sheets but there are formulas that can be used to work this out. Ultimately, as gap becomes much more significant, that formula becomes the one at the top of this answer.

For an ungapped toroid: -

You can see it bears strong similarities with the solenoid formula but "mu" is the permeability of the core material and not air (or vacuum).

Pretty toroid pictures stolen for here

This is a good explanation for inductance with an air gap.