To reach the balcony where you shot the sniper earlier, go around to the area between the front doors and the balcony. Notice the section of rough, climbable wall above. Stand beneath it and jump straight up to grab it.
If you're having trouble with this, it may help to position Lara on or just behind the seam between the floor tiles below the spot you want to grab. (This seam is indicate by the dotted line in the screenshot above.) Then jump straight up, holding Action to grab. If Lara doesn't grab at first, just hold Action + Jump and she'll gradually hop her way forward until she's able to catch on.
Scale the climbable wall upward and to the left, as shown by the arrows in the first screenshot, and then drop onto the balcony.
- OR -
If you prefer, you can take this longer route: Go to the largest of the angled wooden ramps, near the middle of the ENTRANCE HALL.
Stand with Lara's back to the ramp, about one square (three walking steps) in front of it. Then backflip onto the slope and immediately jump and grab the ledge above. Or, climb up the back of the ramp and, as you slide down the front, jump to grab the ledge.
Follow the ledge around to the movable block with the flowers on it.
Push it up against the railing (2 pushes).
Stand next to the block and take a running jump to the balcony with the sniper you killed earlier
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