Take off the trim around the edges of the room using a prybar and hammer, to reveal a space between the edges of the floor and the wall. Don't break the trim as you take it out.
Determine which wall is closest to the damaged area of the floor. Push on the cork floor planks closest to that wall, wiggling and loosening them until they disconnect from the adjacent boards and move into the space. Leave the loosened planks sitting in place, but unattached.
Repeat, loosening the planks course by course, leaving them in the same arrangement as they were. Continue until you get to the damaged planks.
Remove the damaged planks. Set the replacement planks in their place, connecting them along their edges.
Put the floor back together, building back toward the wall course by course.
Reinstall the floor trim, using your trim nail gun to attach it to the walls and cover up the gaps by the walls.