It may be prudent to coat the inductor with additional insulation. Kapton tape may be best for 5kV insulation and low µ. but silicone spray may offer several kV/mm of insulation and work too. THe breakdown between windings is most likely distributed so the maximum potential is not seen against adjacent turns. This may add unwanted capacitive coupling and lower the SRF and there may be better products for improving its safety, perhaps heat shrink tubing if it is small axial type.. The best way to find out is test it with a 1MΩ series resistor so any arcing will be <1 mA and not melt any insulation further or cause damage. Get a 25kV ignition coil from a car. and test it. Put 1MΩ in series and test for current. Resistor case must be >>1kV/mm. 1/4 res are rated for 500V . You can connect in series.
I had designed power supplies a long time ago as a 2nd source for Lexmark laser printers up to 25kV using sine resonant oscillators from 12V stepped up in resonant mode.. No spikes, No pulses, no EMI, no shielding, low cost,, about 25 cents for a multi-output transformer using 10k/day. Regulation was simple R divider tree using enough 1/4W R's to provide feedback at low voltage with 5 outputs some with PWM control from +15kV to -10kV , all < 1% regulation over all specs and cost < $25 in OEM qty. but $350 as a field spare part.