Translation:The children, having eaten, have gone to sleep.
this can't possibly be correct. If I am not mistaken, this sentence is an example of present participle and not gerund. Gerunds in french are mostly introduced by "en" and modify the verb. Present participles modify the noun, like in this sentence. So for one, this sentence doesn't belong in "Gerund" and secondly correct translation should be "the kids who ate went to sleep"
I do see your point here. For a change, "les enfants ayant mangé" is exactly the translation of "the children having eaten".
Alternatives would be: "après que les enfants ont mangé,..." or "après avoir mangé,...les enfants..."
what you propose "the kids who ate went to sleep" is "les enfants qui avaient mangé sont allés dormir" - suggesting that those who had not eaten did not go to sleep.
That's the way I'd prefer to say it, though it's a bit different since it leaves out the pronoun that's in the French sentence (not that that is unheard of when translating French to English). I think this sentence is correct in English, it's just awkward-sounding. I almost think it would sound more natural if the second part of the sentence was something that didn't refer to the children again... for example:
"The children having eaten, we went outside."
You never, never, never repeat the subject noun with a pronoun in English. This construction is common in French, but absolutely not in English! You can say, "The children, having eaten, have gone to sleep." It is wrong to put in "they." The commas are needed before and after "having eaten."
A gerund is a present participle used as a noun. "Seeing is believing," for example. Technically, this lesson encompasses other uses of the present participle, mostly modifiers of nouns. In French, you often use the preposition "en" which makes the present participle which follows a gerund. Since you don't usually use the preposition in English, the present participle doesn't translate as a gerund.
I have put infants on occasion for enfants translation. I think it is correct to fail that translation, as it reinforces thinking that they are the same. Duo, as any good instructor, insists we differentiate and learn from the beginning. Regarding girls, I cannot say ... just lucky I think.