Determining capacitor value in a flyback converter
Is there some other way to work out the switching frequency or choose
those capacitor values?

The UC3842 data sheet shows this graph: -

RT and CT in your diagram are here: -

So, remove C106 and test it with a capacitance meter because there's no other way to determine the switching frequency given that the device may be damaged. You might get lucky by removing the MOSFET and seeing if the UC3842's oscillator magically springs back to life of course.
100 kHz is fairly standard for switching frequencies so if you say that RT (R106) is 15 kΩ then C106 (CT) will be about 1 nF