"eux" means 'them' and "leur" means the possessive 'their', so the sentence would have to be altered for leur to make sense (J'aime jouer avec leur chien)
Leur also means "them" when "them" is the indirect object (COI). She reads a book to them. Elle leur lit un livre.
In your sentence, "les" is a direct object, not an indirect object. Take a look at the tips and notes or open this page in your browser: https://www.thoughtco.com/french-object-pronouns-1368886
Can "aime" in this context not be translated to the English word love? If not, how could one say "I love playing with them." in French? Is "adorer" a stronger indication of love than "aimer"?