Learn Farsi: من به سرکار خانم گفتم که ایشان مدیر جدید هستند. - I told Madam that she is the new manager.

This is the core of the deferential T-V system in Farsi. To show respect to a third person (he or she), we use the plural pronoun ایشان and the third-person plural verb هستند. It operates exactly like using a polite 'they' for a single esteemed person.

What does the word سرکار add to خانم here?

سرکار خانم is a highly respectful compound title roughly equivalent to 'Madam' or 'the esteemed lady.' While saying به خانم گفتم (I told the lady) is grammatically correct, adding سرکار elevates the formality. It is widely used in professional environments, often before a surname.

Could ایشان refer to the 'Madam' I am speaking to, meaning I told her that she herself got the job?

No, ایشان strictly refers to a third person (someone else). If you wanted to tell Madam that she got the job (essentially saying 'I told Madam: You are the new manager'), Farsi keeps the pronoun from the direct quote. You would use the formal 'you' and say من به سرکار خانم گفتم که شما مدیر جدید هستید.