Sadly, if the private key is missing, the wallet is gone for good. Dogecoin core stores the private key in the wallet.dat file in the Dogecoin folder, so if you happen to have any backups of your program files or wherever your dogecoin install is, the private key may be in there. If your computer is straight up broken, there may still be a chance to recover it from the drive, but I don't know the circumstances.
Regardless, I wish you all the best