If you're aiming at 10 kV output, you can look around for a HV transformer from a CRT TV or computer display. This way you would't have to wind the transformer by yourself.
As for the drive: you can use some dedicated PWM driver (TL494 for example) or some small Microcontroller (some kind of ATTiny13 or a small MSP430). In addition you would have to find a driver for MOSFETs to obtain high rise and fall times in order to minimize heat dissipation in transistor(s).
As for the frequency: speaking in simple terms the higher the frequency, the smaller the fransformer of a given power throughoutput is. Good example being a 500W transformer. The ordinary toroidal 50 Hz mains transformer of that power would weigh about 4 kg and be of a size comparable to kitchen bowl. On the other hand a SMPS transformer operating at 30 kHz for the same power would be of a size comparable to a pack of cigarettes and would weigh a couple of hundreds of grams at the most.
Another thing is the magnetostriction effect. It's the effect of ferromagnetic material being strained as an effect of magnetic field. If such strain alternates with some frequency it might be audible for humans, which is highly undesired (very annoying buzzing or squeaking like sound). 17 kHz is beyond hearing threshold for humans (excluding some individuals) so the magnetostriction buzzing is not audible.