Daniel 1:21 says that Daniel remained in the Babylonian court until the first year of Cyrus, which as you say, was 539/8BC, the year Cyrus conquered Babylon and became king of Babylon as attested by Ezra 5:13, as well as king of Persia as attested by Ezra 4:5. When did Darius the son of Ahasuerus (Daniel 9:1) , who is Artaxerxes, who comes after Cyrus and Darius according to Ezra 6:14, when did he become king of Babylon ?
I am not sure. If you read the new edition of my book Daniel (www.ipibooks.com), I am forced to speculate somewhat. Bottom line, we do not have a list of the governors of the province of Babylon. Apparently this Darius was a Mede (and certainly was not the later Darius who was Persian and emperor of Persia). I believe he came to govern this key province fairly soon after Cyrus took it, There was a Gubaru who held this office for a little while. Perhaps Darius took the province of Babylon abound 536 BC. The Artaxerxes who ruled Persia after Darius the Persian was obviously someone different from Darius the Mede’s father. He was much younger. By this time, surely the father of Darius the Mede was long dead.