I had been through that, You dont even need to talk with other party's insurance company. Just call your own Insurance company, describe the date time and location of accident, of course they will ask you whole bunch of questions, answers to the best of your knowledge. Give the other party's insurance information.
As its been a while and you are dealing with another party's insurance company, tell your insurance company that as well.
In fact, you should have had done that on first place. Now they will deal with other insurance company on your behalf. If you have comprehensive and collision coverage, you can ask them to authorize repair, which they will do after doing appraisal. Your Insurance company will pay for repair or they will make another insurance company pay for that.
In either case, its not your responsibility to deal with other insurance company if you were not on fault and other party admitted fault.
Good luck