I might add that there is no other dispute and I was an exemplary client in always paying my account promptly.
The last line would have been my first question, and is an important element to your problem. Very often the only leverage a mechanic (or engineer, as you will) has for payment is retention of the aircraft or logs...and logs are easier to secure than aircraft. In your case, there's no valid reason for the firm to keep them. If indeed they've lost them, then you've got a dillema. While you may have statements showing conformity with applicable airworthiness directives, service bulletins, etc, you still don't have the record of maintenance. This can seriously devalue the airframe and powerplant when it comes time to sell.
You're probably in a position where you're going to need to take legal action, and you may very well be in a position to seek monetary damages for any loss of value to your aircraft. You're best advised to consult an attorney who deals with aircraft and aviation law.