Climb to the bottom of the vine for the longest possible swing. Turn to face the square pillars and aim for the second opening from the left. Swing as high as possible and jump to land on the narrow, uneven ledge just below the square pillars. Then either run to the right along the ledge and jump onto the broken bridge, or climb through the third opening from the left, and the mercenary will not appear. If you go through the first or second opening, you will meet the mercenary.
Alternatively, instead of using the vine, climb onto the corner of the low wall and jump to the ledge below the square pillars then onto the broken bridge. (Aiming for the right end of the ledge seems to work best.) This jump is a bit trickier to land, but it saves even more time.
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