What should my brother do after Hajj, knowing that he swore to leave the Kingdom permanently after Hajj, but then he went back on that? And when is Zakat due on the bank balance to which sums are transferred monthly since 2010, and is Zakat due on all transfers or on the entire balance?

If your brother intended to stay after Hajj before entering ihram, then he must repent. Alternatively, he must offer expiation for breaking his oath not to stay. The expiation for an oath is feeding ten poor persons, or clothing them, or emancipating a slave. If he is unable to do any of these, then he must fast for three consecutive days. However, if he intended to travel after Hajj but a need arose for him to stay, then there is no blame upon him.