Alright, so, extraction.
The best way I can see for getting Hana out is the chuunin exams, it's one of the few ways we can expect to get a specific jounin out of Mist. That, of course, requires Hana to have a team. Luckily, she's a jounin, so if we just say 'Chuunin exam' she'll figure it out. She'll probably be able to play it off as a subconscious need for children to replace her lost son.
So, that'll be the message: the song, and the words chuunin exam. From there, it will be much easier to extract her, since once outside of Mist, she can just cut and run.
Now, how to
get said message to her...I'm thinking a bird. Konoha has summon-birds right? If not, Jiraiya has our platform seal, he can have someone sneak up high over mist. Once they get close enough, they'll genjutsu/control a bird and send it down and have it repeat the tune, and drop a seal holding a verbal message from us.
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Alternatively, if getting a message really is too hard, we use the platform seal, to sneak a nin or 5 over Mist, PMYF down, grab Hana, PMYF up, and GTFO.
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Alternatively, we have Jiraiya grab one of his civilian agents, and train him to perfectly recreate the song. Then, once he gets into Mist, have him 'Practice his flute' somewhere Hazou's mom will hear. When she finds the guy, have him chat with her about what it's like being a ninja and sperg about the chuunin exam. Have him talk about how cool it is that ninja get together and fight it out in a different village twice a year. If she asks about the flute, he'll talk about picking it up on the road from a guy.