Other Issues: Sponsor Problem

So my girlfriend is Canadian and we're trying to change that. Her birth mother is a US citizen, however she went to go live in Canada for a little while before moving back to the states, but came back under a different name apparently to avoid paying something. I don't know the full details of that but that's the gist of it. We have the money, we have 95% of the documents needed to make her a resident or give her a green card...but apparently one of the important things is that she needs a sponsor and it needs to be a relative? There's only one relative of hers living in the states and that's her mother who's practically living off the grid. I can get into more specifics later when I discuss it with her again but I'm hoping someone has some sort of idea of what I'm talking about. I know marriage/engagement is another option to get her over here but I'm trying to avoid it for now if I can.

A few more details:
Her mother can't find her birth certificate to prove she is a citizen, she does have a passport though, and apparently doesn't have to file her taxes because she makes too little. She's a mess. According to someone, she's not eligible to sponsor her own daughter. So...is it possible for, let's say, my family to sponsor her instead? Getting over this issue would fix everything.

Other Issues: Sponsor Problem