Take off the floor trim from the perimeter of the room, using your hammer and prybar. Don't break it as you remove it. Set it aside.
Lay out moisture-block underlayment in overlapping courses over the floor, covering it completely.
Set the first glueless floorboard at one end of the floor, next to your starting wall. Put spacers between the floorboard and the wall.
Snap another board on the end of the first one. Repeat with additional boards, laying them end to end along the whole wall. Put spacers alongside all of them as you go.
Use your miter saw to cut the final board to fit at the end.
Lay the next course alongside the first one, snapping the boards together along their long sides. Arrange the floorboards so the ends of them don't line up between courses.
Repeat and continue, working your way across the whole floor course by course, staggering the ends of the boards as you go.
Length-cut the boards for the last course on your table saw, to fit alongside the ending wall while leaving a space of about ¼ inch there.
Reinstall your floor trim with your trim gun, hiding the spaces between the flooring the walls.