I've also made a short video detailing this sequence.
You can see the key through the grille on the left, but you can't reach it.
Go to the right toward the wall of spikes and locate the switch on the left wall.
Save the game here if possible. When you pull the switch, the grille at the other end of the hallway opens, and the wall of spikes begins moving toward you. Immediately turn left...
...and start sprinting toward the key. Release the Sprint button a few steps before Lara reaches the key. This will prevent you from overshooting it and having to waste time backing up.
Stop directly above the key and press Action to pick it up.
When you take the key, the gate where you came in re-opens.
Roll and start sprinting back the way you came.
When you first start running, the camera will be pointing toward Lara awkwardly. If you like, you can press Look to get your perspective back.
Release the Sprint button just before you reach the doorway on the right. This will give you more control as you veer into the opening.
Wait for the spiked wall to pass.
Then re-enter the room and head to the right end, where the wall of spikes originated, to find MP5 clips and a small medi pack.
NOTE: If you're having an especially difficult time outrunning the spiked wall, try this: Pick up the key then step forward so Lara is standing with her chest against the wall. Wait for the spiked wall to approach, and as soon as it touches Lara, backflip. She'll go through the wall, taking some damage, but if you start with full health, she should survive. You can also use a medi pack while passing through the wall. (The video also shows this technique.) Thanks to Sven and Terry, both of whom submitted variations on this strategy.
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