If wire's feeding, the torch trigger etc. are ok. (does the gas come on too?), it could be the contactor - looks like a very big relay (cos that's what it is), switches mains to the welding transformer - you should hear it click, if it's in a clear plastic casing you can often see the contacts moving too. Check the connections haven't shaken loose *with the power disconnected* too, another check would be the variable current switch for the transformer that picks which mains tap and sets the arc voltage/current.
If it's the contactor, shouldn't be that pricey, get a better replacement from TLC electrical or the like, take the dud in so they can match it for current and coil voltage.
The switch is going to be more difficult, you'd need a mains rated multiway one, with a high current rating (eg 25amps) - possibly simplest to get one from the maker/importer if so. If it's using a bunch of rocker switches check whether *none* of the settings work, that ,suggests no power getting to the switches, again the contactor or a loose wire could cause that.
Dave H. (the other one)
Took a closer look at the pic...
The white sleeved wired from the switch are the "out" side, on the switch there should be the power coming in too - that should be coming through the contactor, the circuit board at the top rear should supply the *coil* voltage for it, so follow the cables - there should be one each for:;
Wire feed motor,
Control transformer (not the huge one, or the choke connected to one of the welding leads, one about the size of you fist),
The contractor feeding the power selector switch.