Emily's mother is ironing seemingly to prepare for the next day: She has been asked to come to school to talk about Emily and Emily has mid-term exams. She is also ironing because in 1961, people still ironed clothing as a matter of course. Ironing is a process that smooths out wrinkles and puts creases, such as pleats, in where they belong. It is basically a process that manages and regulates fabric that has become unruly.
Emily's mother has been asked to make time to come to school because Emily is "a youngster who needs help," a youngster who needs managing. In this sense, the iron and the dress are symbolic of Emily's troubled past and need of managing. The iron then represents those who try to manage her.
Emily's mother is not included in the iron symbol. For one thing, she is contemplating the request for her time and the unidentified individual who made the request, which separates her from those who wish to manage Emily. For another thing, she describes herself as encouraging Emily and trying, if perhaps failing, to breach the separation between them.
In the end, after Emily's mother's contemplation of managing an unruly young woman through the process of managing unruly fabric, she comes to the conclusion that Emily doesn't need managing because she is more than an unruly outer shell of a person. Emily's mother concludes that with greater help to know herself, she will see that she is more than a empty garment that has become crumpled or unruly.