Please again note that historically Cyrus does not begin his rule the same year Babylon falls. For instance, Sir Isaac Newton in his "A Short Chronicle" where he lays out the NB timeline, he allows Darius the Mede a 2-year rule before Cyrus comes to the throne and releases the Jews. So the Jews are still in exile during the reign of Darius the Mede. Further, the context historically for Darius the Mede is that he is grandson of Nebuchadnezzar and thus a legitimate Babylonian king...
Here's the exerpt from that reference...
604. Nabopolassar dies, and is succeeded by his Son Nebuchadnezzar, who had already Reigned two years with his father.
600. Darius the Mede, the son of Cyaxeres, is born.
599. Cyrus is born of Mandane, the Sister of Cyaxeres, and daughter of Astyages.
596. Susiana and Elam conquered by Nebuchadnezzar. Caranus and Perdiccas fly from Phidon, and found the Kingdom of Macedon. Phidon introduces Weighs and Measures, and the Coining of Silver Money.
590. Cyaxeres makes war upon Alyattes King of Lydia.
588. The Temple of Solomon is burnt by Nebuchadnezzar. The Messenians being conquired, fly into Sicily, and build Messana.
585. In the sixth year of the Lydian war, a total Eclipse of the Sun, predicted by Thales, May the 28th, puts an end to a Battel between the Medes and Lydians: Whereupon they make Peace, and ratify it by a marriage between Darius Medus the son of Cyaxeres, and Ariene the daughter of Alyattes.
584. Phidon presides in the 49th Olympiad.
580. Phidon is overthrown. Two men chosen by lot, out of the city Elis, to preside in the Olympic Games.
572 Draco is Archon of the Athenians, and makes laws for them.
568. The Amphictious make war upon the Cirrheans, by the advice of Solon, and take Cirrha. Clifthenes, Alcmaeon and Eurolicus commanded the forces of the Amphictions, and were contemporary to Phidon. For Leocides the son of Phidon, and Megacles the son of Alcmaeon, at one and the same time, courted Agarista the daughter of Clifthenes.
569. Nebuchadnezzar invades Egypt. Darius the Mede Reigns.
562. Solon, being Archon of the Athenians, makes laws for them.
557. Periander dies, and Corinth becomes free from Tyrants.
555. Nabonadius Reigns at Babylon. His Mother Nitocris adorns and fortifies that City.
550. Pisistratus becomes Tyrant at Athens. The Conference between Croesus and Solon.
549. Solon dies, Hegestratus being Archon of Athens.
544. Sardes is taken by Cyrus. Darius the Mede recoins the Lydian money into Darics.
538. Babylon is taken by Cyrus.
536. Cyrus overcomes Darius the Mede, and translates the Empire to the Persians. The Jews return from Captivity, and found the second Temple.
529. Cyrus dies. Cambyses Reigns,
From "A Short Chronicle" by Sir Isaac Newton....
Again, there is no problem with the end of the 70 years occurring after the rule of Darius the Mede since the Bible is specific that it is when the "royalty of PERSIA begins to rule" that the 70 years ends. This explains why Darius is specifically called Darius the MEDE and Cyrus, the PERSIAN. The "royalty of Persia" is thus a reference to when Cyrus later comes to rule over Babylon and does not include Darius the Mede, who is not only Persian but half Babylonian and a son of Nebuchadnezzar. Darius the Mede, historically served as a captain in the Babylonian Army, a position customary for princes of the empire, etc.
So there are some technical complications. The Jews were clearly in exile during the reign of Darius the Mede. Otherwise, Daniel would have left, right? He was still dealing with the jealous priests during the reign of Darius the Mede. Only when Cyrus came to the throne and released the Jews and many others did these 70 years end.