How do I measure the Inductance
The obvious starting point would be to count (or estimate as accurately as possible) the total number of turns, length, and average diameter. Then plug them into the standard equation:

L = N2r2 / (228r + 254len)

Where:

N = number of turns
r = radius of coil in meters
len = length of coil (also in meters)

...and the result will be in millihenries.

Oh, and since it's apparently operating at RF, you'll need to de-rate that by a couple of percent or so to compensate for skin effect in the wire.

I think the intent is that the resistor only acts as a physical support, without affecting its inductance much, but I could be wrong about that. If you want to measure, you could build an LC tank circuit with a capacitor of known value, measure the resonant frequency, and compute the inductance from the normal equation, fr = 1/[2π√(LC)]