I promptly resumed the former position of success.
Because you have to stop and think a little bit about this sentence, it is a humorous way of saying, “I sat on my haunches again.”
promptly = immediately
resume = begin to do something again after a pause or break
the former position = the position before (in this sentence, it means the position before the author stood up)
of success = Because Jesse was not crying when the author was on her haunches, this was the position of success. It was a position that was successful in keeping Jesse from crying.
to sit on your haunches = to squat so that the back of your feet (your heels) touch the back of your legs
See STORY example