You need to contact your insurance agency as soon as possible.
Yes, you can challenge her in court and demand proof and receipts of all damage you have done and exactly what she is charging you for.
She may be asking you to pay for an entire new fence and repatching her entire lawn. That's the only reasons I can come up with for why it's come to that amount of money.
It really is insane.
You may also want to hire a lawyer.
In a criminal case there are 2 attorneys involved: 1 representing the Defendant (the person charged with the crime) and 2. the prosecutor/or district attorney (same thing just called different things in different states). The prosecutor works for the government, usually a county, and pursues the case on the states behalf. Witnesses and victims may have attorneys, but they take no part in the proceedings.
NO lawyers can go from side to side, not even in civil litigation, that would be called a conflict of interest (which I will not go into defining because you asked that we keep it simple ;)