How is it you have a recommendation from a judge if you did not already sue for the damages to your dog - was this in an action brought by the police against the dog's owner?
Document all your expenses, send COPIES of them as part of a demand letter and, when the owner does not pay, sue him/her in small claims court.
The foregoing is for general information purposes and does not establish an attorney-client relationship.
You can file a claim in Small Claims Court for any case valued under $2,500. Small Claims is designed to be friendly to the public and you should be able to navigate the system without an attorney.
Generally speaking, a court can award a judgment. It can also provide other relief, like an order to remove the attacking dog, but this would be pretty rare, especially in Small Claims Court. More typically, the result is a money judgment.
Disclaimer: This post is for informational purposes only, and is not intended to provide legal advice or establish an attorney/client relationship. Legal advice should obtained by discussing the specific facts of your case with an attorney licensed to practice in your area.