Backpay is simply the term for wages
in the past. The EEOC is saying that you should pay her for 40 hours worth of work (in the past), which is called backpay.
In regard to the bias information, I think you can safely assume that the EEOC is biased. They are an agency which is established to be advocates for employees. However, you must remember that the EEOC finding has no actual effect other than to simply give you an idea of what an advocate against you believes you owe. In other words, there is nothing to appeal at this point. The next step is the "conciliation" process where the EEOC tries to get you to settle with the woman.
It is only where the EEOC files suit that you can really take any action to combat them.
If the EEOC won't conduct a mediation, then I can almost guarantee you that they will not sue. There is case law out there which would force the EEOC to drop the case and go back to mediation, as the EEOC has a statutory duty to try to conciliate the case before they file suit. Sending you a letter and demanding that you pay without giving you a chance to mediate is hardly meeting that duty.
So, in this case, I think you should go ahead and make your offers to the woman. If she gets a right to sue letter and then retains an attorney, you should go ahead and get an attorney to defend you.
What she has against you is a facially valid claim because of the things that were said to her. However, you can explain what was said and can also show that you were not hiring at that time. If you are not hiring, you are not hiring...its a valid defense and it wins.
I understand you are at a loss as to why this is going on, because it is not fair. But this is a process and unfortunately, you have to jump through the hoops and probably throw some money at the problem to make it go away.