Advice: there is no legal liability for this accident, but perhaps there is still a way to recover compensation.
There is no strict liability. There was no leash law violation or other animal control violation. Nobody had a dangerous dog. In short, nobody did anything wrong.
For that reason, there is no legal liability. Our society requires grounds for legal liability. If there are no grounds, there is no legal liability. Either you answered a question wrong, or you cannot make a legal claim against the other party.
However, there may be other considerations which would motivate the other dog owner to pay something to you. People are moved by ideals and moral obligations. These include neighborliness, family harmony, and empathy for a fellow dog owner. The Preface of Volume 1 has powerful information about how people feel regarding these matters.
Since you do not have legal grounds, examine whether you have other grounds. If you do, you can consider appealing to the other dog owner — but not in court, and not by making an insurance claim. Court cases and insurance claims require legal grounds.
It is recommended that you go through this Liability Wizard again, to make sure that you didn’t miss anything.
After that, you can read Volume 1: Text and browse Volume 3: Forms to get more ideas.