are you going to argue you thought the lawyers were doing this all for free? Unless so, you are going to have to come up with some understanding as to the payment agreed upon. And no, you are not going to escape paying just because a written contract is not producible. In many situations, a verbal contract is quite enforceable. The terms would be the only arguing point.
bottom line; you will pay them something. if you want to argue it, you will find yourself in court, against a lawyer. You may end up hiring another lawyer to defend yourself against this one. I have no idea how much the current attorney is seeking but it could be quite costly to defend yourself against him and you will end up paying him something.

