So from what I can tell: (And obvious Spoilers for key location etc.)
1 Don't move.
2 Reach Trowzer without the key after the time limit.
3 Reach Trowzer with the key, but after the time limit (you can grab the key later. It's Upwards, first left, past the minefield, then up until you are in a room with four exits, all but the bottom one surrounded by 2-wide lines of mines. The key is in one of the three adjacant rooms, but it's rng which one)
4 Get the key and reach Trowzer in time
5: Collect all keys (including the one you have to collect as the first-it's in one of the rng rooms, apparantly)
6 (Edit: Thanks to Garry_T_Fox) Collect the dead-looking key in one of the rng rooms. You CANNOT collect any other key before it. Return to Trowzer's room and go right, past Shovel Knight until you find a big key at the end. Bring it to Trowzer.