I got past that part, but I could not get past the gap right after. I could jump until I got close enough to the edge that I felt like I could make the jump.
You aren't supposed to jump over there. After the part you said you did where you went to the new door, then you walk to the left, jump up and go through the second gate. Then walk to the right, jump in the new patch of vines, and it will take you to the exit door.
