EDIT 2: After more research, it is looking very possible that this is a glitch. You cannot flush the sword right away in Normal mode or Mary's mode. It seems to do it because you have the broken sword. Its possible that it is still by design, but I wouldn't wait too long if you want to do it this way.
I have left a video showing the path below along with my original video.So use whichever method you would like!
EDIT: It turns out you don't need to beat the game fully to unlock this. You simply need to make it to the toilet to flush the sword. So a quick path would be to get the sword thrower, gardening glove and the hotel guest over the bridge. this will allow you to run from the hotel to the factory toilet and get your achievement. I'll leave the original guide below for anyone who wants to beat the game fully in this mode.
ORIGINAL: To unlock Second Run you first must beat the game once. After that, it will be an option in New Game +.
Second Run is essentially Hard Mode. You only get 1 heart, with no way to upgrade it, and you only get 40 seconds per run instead of 60.
Certain items are moved around as well to increase difficulty.
Here's a video of me beating Second Run mode, showing you the path I took to get it done in under 30 minutes. I have edited out unnecessary deaths. However, even with those deaths, the final time was less than 30 minutes.
If the achievement doesn't unlock (like in the video) hit continue and run to the boss and kill him again. I show the path to get there from the starting house after I clear it for this reason.
Full Playthrough with Boss Kill: