Aw, I read this and felt sorry for the delusional person. I’m just going to assume that she is embarrassed that she is only a nursing assistant (when really, that’s nothing to be ashamed of!).
The whole situation is weird though and I think she should just do what your gut tells you. To be honest my first response was; If you can just let it go. But that’s mostly because I felt sorry for the girl.
Anyway, because it’s obviously bothering you I’d say you should either think of a way to “catch her in her lie” Or, just simply bring it up. You can lie about how you found out, so it doesn’t look like you were sneaking around behind her back (not that you were, but you know what I mean..!). She’ll obviously be embarrassed, and probably defensive…and it won’t be a happy conversation…but you definitely need to do something about it. Just try to be sensitve and not angry when approaching her!