I was encountering the same problem with the "Broken Steel" quest not staring up after continually dying in that room where you are to enter the code. I found out that if you enter the code correctly before you pass out, that it would begin. So, as a tip folks, go to the room and enter the correct code as fast as you can. You'll still pass out, you'll see the Jefferson statue as your vision starts to fade to white, the ending story with the narration, then you'll see "two weeks later" materialize on the screen immediately afterwards. Hope this helps. :)