"Sie liest mit ihrem Kind." shouldn't be translated as "She reads with your kid", am I wrong? "Ihrem" should begin with capital letter, I hope!
mit ihrem Kind = with her/their child
mit Ihrem Kind = with your child
Wouldn't "She is reading with her child" be the most natural reading of this?
The way I explain myself the construction of the phrase is like this:
her -> ihr (Genitiv to express possession)
mit demands Dativ, so the whole particle "her child" must be Dativ
Dativ for neutrum (das Kind) demands "em" (dem, der, dem, den)
Another pronoun that would give the same result (ihrem) is "their", as explained by Katherle.
This logic worked for the exercises I encountered until now in this lesson, however it would be better if someone who knows German confirmed this.