How can I pay the zakat on money that my father deposited into my account without my knowledge of the deposit date, its original amount, or how he disposed of it, given that I only know its current value and that two and a half years have passed since his death?

You request data from the bank regarding the amount and its transactions to calculate zakat (definition) at a rate of 2.5% for each lunar year in which the nisab (definition) was reached. The zakat paid is then deducted from the principal and its profits for the following year. If obtaining the data is difficult, you should endeavor to estimate the amount, its profits, and the time it reached the nisab, so as to give what you predominantly believe will clear your liability.