What did Middleton do to free the white hostages from Big Bear?
The brief answer to this is that Middleton used military force to get Big Bear and the rest of the Cree to release their hostages. This happened at the Battle of Loon Lake, also known as the Battle of Steele Narrows. This was the last battle of the North-West Rebellion.
In this battle, the Cree band made a stand against troops led by Major Sam Steele (Middleton was in overall command, but was not present at this battle). Middleton had told Big Bear that he would pursue and destroy the Cree band if they did not release their hostages. That is what Steele was doing. When the Cree ran low on ammunition, they released their hostages and fled.
Middleton was only able to free the hostages, then, by military force.