From personal experience... this is the long way to go about it.
The shortest route is to cut your own insurer out of the loop, they only serve to introduce unnecessary delays and tend to obfuscate matters.
Contact the other party's insurer directly. They will contact their client to get his/her side of the story. If the other-party's insurer accepts liability (either because their client admitted they were at fault, or because you have submitted overwhelming evidence), then they will be very eager to settle the claim as soon as possible. They will know very well that if you contact an accident management firm, the cost to them could double and more.
The above won't work though in the event that the other party denies liability, and you have no evidence to refute their claims. In this case, you either drop the claim altogether and take it on the chin, or submit a claim on your own policy and let your insurer battle with the other party's insurer. You can also take the other-party's insurer to court yourself, but this is not generally a recommended course of action.
But again, if the other party's insurer accept liability... they will be chasing you for a quick settlement, before their own costs start to balloon.