Is the expiation of liability for stolen money—given the impossibility of identifying its owners or precisely estimating its value—achieved by donating it on their behalf, and will repentance and good deeds be accepted thereafter?

Your repentance is sound and accepted. Disregard the whispers of Satan. As for the money you wrongfully acquired, it must be returned to its known owners, even if indirectly and without informing them. It is not permissible to give it as charity if it is possible to deliver it to them. However, for those whose identities you do not know or whom you despair of reaching, you may give it as charity on their behalf. Turn to your Lord with good deeds, and be certain of the acceptance of your repentance and the forgiveness of your sins.