Your children’s current status are irrelevant at this stage. The reason for her departure while AOS was pending (even though she shouldn’t have left because she was effectively considered to be in an authorized stay because of the pending AOS) is also irrelevant at this point, I’m sorry to say. If truly considered about overstaying illegally, she shouldn’t have entered the US on a visitor’s visa to file for AOS. Anyways, that’s all water under the bridge now.
Like I said earlier, her GC was approved in error because USCIS didn’t realize she already left the country a couple of months before it was approved. Going to the embassy, assuming you can get an appointment to enter the premises, to “explain her case” isn’t going to help her situation. Her GC, technically speaking, is no longer valid.
Now, she can decide to take her chance and travel down to the US with the card. She may be re-admitted, slight possibility of that, however there’s still the fact that the GC is technically speaking no longer valid. I do not believe she will be able to renew the card or apply for citizenship, years down the road. USCIS will review the entire immigration history then and they will discover the GC shouldn’t have been approved in the first place.