How did Iroh deliver the "Secret History" message to Zuko while imprisoned?

Maybe the lady guard who was very polite and kind to Iroh gave it to Zuko

Iroh was a suave and well-connected guy [see: entire series]. He probably just asked a friend to do it. As you mentioned, he had allies in the White Lotus, who were quite sly and tended to be conveniently placed. Then there's his female guard, who he seemed to have developed a rapport with. Any of them could have reasonably had or gotten access to Zuko's quarters on Iroh's behalf. There don't seem to be any solid clues in the episode as to how it actually wound up there, but after all it was just a letter delivery.

In the real world, prisoners - even political prisoners like Iroh - tend to have access to correspondence with the outside world. Think of Nelson Mandela, or the Apostle Paul. It's not much of a stretch to assume that Iroh was simply allowed to write letters, and asked someone he trusted to deliver it for him.

I was thinking it could be either sokka’s master because he’s part of the white lotus and lives in the fire nation, or it was king bumi who was also a member of the white lotus, had the same crouched stature of the cloaked figure, had the same weighted footsteps, and Bumi could have been hiding in plain sight under the protection of the white lotus.

It couldn’t have been Bumi because Bumi was still a prisoner in Omashu until the Day of Black Sun when he finally escaped. More than likely, he just had someone deliver the note to him.

I’m pretty sure it was the lady guard because yes it would make sense for Sokka’s master to deliver it since he is in the white lotus and is in the fire nation, but how would he know that Iroh needed to deliver a letter to Zuko? Iroh had no way to tell Sokka’s master about the letter, so I definitely think it was the lady guard.