If non-kosher food is eaten accidentally, do you have to do anything afterward? Is some form of “purification” required? If so, what?
"Purification?" No. (And this kind of thinking has unfortunately lead to OCD in some people.)
Technically, if I eat a non-kosher-slaughtered chicken, that renders me "ritually impure", and I can't enter the Temple until I do a ritual bath and wait until nightfall; but those laws are generally moot with regards to the world in which we live today.
Eating non-kosher is a sin like any other sin, and follows the same atonement process. In Temple times, some severe categories of non-kosher would require a "mistaken sin" sacrifice; today we don't have sacrifices. Basically, it's feel remorse that you did it (which if it was really a mistake, you probably already have remorse) and make plans to educate yourself better so you don't make such mistakes again.
Lastly, for trivia's sake: technically, human milk is always kosher, regardless of what the human was eating; however, some have a fixation about if the mom ate non-kosher. So if a nursing Hassidic mom ate non-kosher by mistake, her rabbi might advise her to pump and dump for a day or so until it's out of her system. (Again -- the formal law doesn't require this.)