There is another spring that goes on the handle side of the bar between the door handle and latch. Your spring remnants could be from that too. I just replace all four springs on my doors. See this for a picture

https://www.dropbox.com/sh/mky62i9ciq1gnqx/AABCx5GDEuoujQcqd0H1c3fPa?dl=0

David