Basically, no, you can't find out. Airline won't tell you because of privacy concerns. Even if you take her to the airport and watch her go through to immigration, she can still go back once you've gone.
There is no point in pursuing this. After you fire her, send Imm D the relevant form. Then she is no longer your concern or responsibility . At best you can write her name on the black list here or even write another letter to the Imm D.