To answer your last question first, no, Odin isn't the only way to wipe and install new software, but it's pretty much the only way if your phone isn't already rooted.
You could use something like Mobile ODIN:
http://market.android.com/details?id=eu.chainfire.mobileodin.pro (your phone must be rooted with the bootloader unlocked to use this), or if you have an alternate Recovery installed (like TWRP or ClockworkMod) then you can just boot into Recovery to do that.
Is it accurate to say that she had been running this 4.2.2 ROM for awhile, with WiFi working normally, but yesterday she did a Factory Reset and now WiFi no longer works? Or did she load the ROM and do a Factory Reset (as part of the load process) and then WiFi no longer worked?
It's just strange that a Factory Reset would have messed up a functional WiFi, since the radios aren't affected by a Factory Reset.
Unfortunately I'm not familiar with the various carrier-specific flavors of ROMs, so I can't give you any specific advice. You could try reloading the new ROM and see what happens, or just revert back to the original ROM and make sure that everything works.
What was the impetus for switching to a different ROM?
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