If you are not in a no-fault state. If you are, your insurance company should be suing (formality) the other company to get your deductible back. You do have a police report I hope. The police report should say who is at fault.
Insurance investigator told me, after an accident, that they often do not even read the Police reports. They do their own investigations and determine fault on their own..............
Yeah, op said fault admission was in the police report.
BTW, this is why I have a dash cam.
Just an an FYI, no-fault almost always ONLY applies to injuries, not property damage.