Oops, forgot the component values. Have updated the image in the OP with them.
I tried your idea, and it does seem to help a bit, although the coil's output is no where near as powerful as it used to be. I tried replacing the 10K series with like 3.9K and it didn't really help. Think it might be the PLL having trouble locking on as I've had it happen before.
I have also tried reducing the track length between C10 and the 4046 which didn't seem to help.
As a side note, I did a frequency analysis of the whine last night, and it has fundamentals at 775Hz, 1549Hz, 2323Hz, 3882Hz and some other inaudible frequencies.
I have a feeling that this may actually be coming from the PWM used for adjusting the power (Using P1, a 10K potentiometer)
Is there anything on that circuit that would actually generate a frequency around those ranges? It's possible it could also be a beat frequency, but it's pretty difficult to analyze.