What other means are available to reach the owner of the damaged car and ask for his forgiveness, other than trying to search for him in newspapers and performing the Istikhara prayer?

As long as you did not intentionally do what you did, there is no sin upon you, nor is there any blame if the car is restored to its original condition through repair at your father's expense, for Allah, the Almighty, says: "And there is no blame upon you for that in which you have erred, but [only for] what your hearts intended." What is required by Islamic law is to remove the damage and repair the defect that befell the car. As for its lost earnings (ghalla), you are not obligated to compensate for them because you are neither a usurper nor did you act intentionally. This is a normal error, and the one who made the mistake is obligated to rectify what he spoiled. If you pay something to the car owner to appease him, that would be better.