DC/DC converter from 9V to around 180V - LTspice simulation
You can improve your model by placing the resistance of L1 in series with the inductance.  But your real issue is that the BSS123 is rated to only 100 volts and low power, so you will need a higher voltage one.  I would aim for about 400 volts as a minimum because your coil will have some high voltage spiking.  You need a FET capable of switching at least a few amps so you can't have an SOT-23 package - go with a TO220 or TO263.  Good luck!