If you believe religion is a private matter, then your answer might be something like, “He prayed, read the Torah at home, and maybe started a small group with Shadrach, Meshach, and Abednego.” All good possibilities, but isn’t it also possible Daniel attended some sort of worship gathering?
Judaism was designed to be a national religion of a settled people. They would have one center of worship, and a priesthood would be available for everyone. The Babylonian captivity made this impossible. The temple was in ruins and its site far away. Sacrificial worship and religious festivals were impossible. How would they worship?
The Jews were dominated by the Babylonians, and succeeding empires. They were forced to interact with new cultures and religions. Often they were forced to worship other Gods, as we see in Daniel 3. How could the Jews preserve their faith and resist incorporating other gods?
During the exile, a new form of worship developed in the absence of the Temple: the synagogue. There Jews could pray and receive instruction from the Torah. Over time the synagogue became a permanent form of worship, even after the Jews returned to their homeland during the Persian empire.
The establishment of the synagogue was a movement to preserve a pure remnant of Israel, a people dedicated to God and His Law. If there was one around in his day (which is entirely possible), it’s hard to imagine Daniel sleeping in on the Sabbath — unless Nebuchadnezzar wouldn’t give him the morning off.
Aside from the history lesson, there is a two-fold moral here: 1) The worship of God is deeply personal, but never private. 2) Corporate worship is necessary to keep other gods out of our faith.