The metaphor in question can be summed up with the narrator's wish for her daughter in the final line of the story. She hopes that Emily will know "that she is more than this dress on the ironing board, helpless before the iron."
In other words, she hopes that Emily will become more than a conformist, that she will realize that it is she who shapes her life, not some outside, demanding and pressing force. She hopes that despite her less-than-perfect parenting and the harm she fears she has caused her daughter, Emily will discover the beauty that is within her and realize that she does not have to be what the world or her society tells her she must, no matter how much pressure is applied to her to force her to conform; Emily has the power within her to decide her own future and is not bound by her current or past circumstances--she is not "helpless before the iron," but rather, has the potential to shape her own life. The narrator only hopes that Emily sees that power within herself.