Is there a difference between the word "shows" and "mirror"?
No, they're the same. Feminine singular present tense of להראות - to show - is מראה, mar'ah, just like the word for mirror.
This should really be translated as "the boy shows the book to the girl", not "the child shows..."
Dov, this is an issue I've been flagging often as well! While it is possible to translate ילד as "child" in some contexts, the gender has to be unspecified/ irrelevant. The moment they try to contrast it with ילדה, though, ילד can only mean "boy."