"Leka" in Swedish is in terms of the "children playing". Yeah, many different specific verbs for different kinds of contexts. "Spela" is more general but in this context, "Leka" is in terms of children.
Shouldn't the English version be "...the whole day" then? And why is it not "... leka den hela dagen"? Isn't hela an adjective for day and therefore dag ahould have both the article den preceding it and the -en ending?