You need feedback from the HT generator to detect both sparking and to regulate/set the HT.
I'd probably have gotten higher than 170V before going to the bother of designing a case.
Designing HT supplies is non-trivial, you'll probably want to pot things in clear silicone/RTVwhen you get higher. Also, when a discharge happens you can easily nail the FET/transistor, so look at adding protection in the form of over-current and over-voltage.
If you are going single polarity then LOPT's might be better to start with, they are designed to work around 16KHz and have the output rectifier diode(s) built in and are designed to drive a capacitive load (eg CRT).
Also, if it's contact only, then I'd be tempted to just use the relay to discharge/safe the probe-tip (via a few hundred K) when the trigger is released.