How is the Zakat calculated on a sum deposited in an Islamic bank after it has reached the minimum threshold (nisab) and a full year (hawl) has passed, and is the Zakat paid from the principal amount or from other sources?

If your wealth reached the nisab (the minimum threshold for zakat) last Shawwal, then its zakat becomes due next Shawwal, and it is permissible to pay it from that wealth or from other wealth. Any additional wealth is not subject to zakat unless a full hawl (lunar year) has passed over it. Its nisab is the same as the nisab of the initial wealth, unless it is a product of the initial wealth, in which case all of it is subject to zakat next Shawwal.