What is the ruling of Sharia on money taken by children from their father's wealth without his knowledge, with the intention of repayment, when the father died without being informed or the amount being repaid, and should the amount be given out as a perpetual charity, or inherited by the heirs, including the children, or given to the mother?

You both have committed a grave error, and what is required is sincere repentance. Among the conditions for repentance is returning rights to their owners. This money has become the property of the heirs after your father's death. Therefore, it is obligatory to pay each heir their share. Your consciences will not be cleared except by doing so. It is not permissible to make it a charity except with the permission of the heirs. It is not a condition to inform the heirs of how you took the money. Fulfilling this debt takes precedence over performing ongoing charities.