This sounds really sad to me, I hope you guys can find a solution

Okay, I don't know my stuff. But just possible options in my head...what if you have your wife's godmother legally adopt your wife as a daughter? Does anyone know if that'd be enough.
I don't know your wife's situation but it just sounds like they're really close, and if she basically raised her then maybe your wife doesn't really have her real parents around...? Sorry, that's a leap, but if that's true then maybe adoption would be viable. I don't know if the government considers that 'family' then, though.
The other thing I'm thinking of is any sort of work visas the godmother could get. It's a good help, I would think, that she'll have a support system when she comes into the US, a place to live sounds like, etc. She doesn't have any 'specialized skill' but really so long as she's not going to be a burden on the economy I don't see why the government would care.
Even if you could only get some sort of temporary work visas or clearances (which sorry, again, I don't know what your options would be here), and just have them renewed as much as possible, maybe this would be okay...? This is REALLY insensitive, but, how old is the godmother? It sounds like she's older to me, so maybe a plan that would last however many years but not make her a permanent citizen would be good enough...
I'd say, if you do find an option that you think gives you half a shot, to go for it. Fill out all the forms and if I were you I'd write lots of notes throughout all the paperwork explaining the special circumstances, explaining that you'll fully support her, etc. I mean the worst that can happen is they turn you down. If you can afford the filing fee then I'd go for it.
Really really, good luck!
EDIT:
You know what, I think I read this completely wrong

I was reading it like the woman is your wife's godmother, not your daughter's. Looking at it now I think you meant your daughter's, oi. I'm really sorry.
Well, if she's a younger and perfectly capable woman, then go for the work visa I say, or just get a really good frequent flyer membership