Who regulates the balancing of the books for commercial banks? How? Is it public information?
It's public information, and you can take courses, and even university degrees in it.

The double entry book keeping technology used by banks is essentially designed to tackle exactly this problem, and all banks and commercial customers use it. The system itself provides a single error detection/correction mechanism, that guards against human error, and also originally required updates in two separate books, maintained by two separate clerks - which was deliberately done to make fraud more difficult. Essentially if the books don't balance, this is a red flag to find the mistake/fraud that caused this to occur and correct it. These days it's all computerised, but the underlying operations are identical.

The entire accountancy profession's job, is essentially to supervise this process, and periodically audit both banks and companies in order to make sure that the books are balanced, and that nothing illegal is being done.