BioShock

You could go to youtube and check out a player's runthrough on there and then emulate exactly how they did it. Plus, I'm sure with a bit of practice after watching said video, you could run through it with your eyes closed. Personally I haven't done that challenge yet, but I know that I've seen videos for every possible aspect of the dlc levels. Hopes this somewhat helps.

This is a good series of videos to reference: http://www.youtube.com/watch?v=fPC2KX6A2fc
Note how fast he does it, and benchmark your progress through the levels. If you're significantly slower than the times he's getting, reload your last save and try it again until you're faster. Playing through the whole Worlds of Hurt level from start to finish over and over can get very taxing, but simply playing the individual parts over until you've mastered them isn't so bad. Keep in mind that in order to beat it in under 15 min, there's some luck involved! Minimize the luck factor by "locking in" good runthroughs of individual rooms using your saves. Good luck!

YOUR TALKING ABOUT 2 DIFFERENT TROPHIES. PASSING IT WITHOUT WEAPONS EXCEPT THE WRENCH IS A SILVER TROPHY AND PASSING IT IN UNDER 15 MINUTES IS A WHOLE DIFFERENT ONE.
1.PASSING IT WITH WRENCH- TRY GETTING SECURITY BULLS EYE TO THROW TO THE BIG DADDIES IN TH ANNOYING BOT LEVELS. GET THE BEE SWARM 3 TO KEEP BIG DADDIES BUSY. GET THE ENRAGE PLASMID TO THROW AT OTHER GUYS(IN A GROUP NOT BY THEMSELVES). its A LOT EASIER TO GET BIG DADDIES FIGHTING EACH OTHER.
2. UNDER 15:00- TRY RUNNING THROUGH IT WITHOUT COLLECTING EVERYTHING

Best guide for all of the dlc challenge rooms is by 1st_level_grunt on this site, print it, and follow it to the T you'll beat every room. I did. : )