J-1 visa is max 2 years. If she is not older than 26 after returning home and waiting 2 years, than she can reapply for a new J-1.
There is no extension after completing the 2nd year on the same J-1 visa. No exception.
She can try to get a tourist visa but cannot legally work.
I know of an AP who completed her 2 years as an AP and successfully got an F-1 and is now a legit graduate student at a university. She is not working for any family in a childcare capacity. Working (legally) under an F-1 is very restrictive.
Work visas are for people who have qualifying skills. Caring for children is not one of them.
AP can be a domestic employee or nanny on a B-1 visa but she must be accompanying a foreign national employer (you). Since you are a US host family, you are likely not a foreign national as well - not sure how dual citizenship would play here.
https://travel.state.gov/content/travel/en/us-visas/visa-information-resources/all-visa-categories.html
This has been discussed many times. If there was a (legal) magic bullet for an AP to stay beyond year 2, you would have found it mentioned all over the internet.