To make a circle in the middle of that inset area, I'd select that one polygon, inset it until it was about the size of your cylinder, add a horizontal and a vertical loop cut (making the inset 4 rectangles and 8 vertices), edit everything to make it smooth again (the loop cuts will make them more blocky), then delete the 4 inset rectangles to make a hole.  Then I'd use the Loop Tools circle or something like it to make my rectangular hole into a circle.  Then I'd just extrude the circle out into a cylinder, cap it off (avoiding tris), then add loop cuts to the top and bottom to give the cylinder cleaner edges.