Akiko and Zuko get ready for bed

After their betrothal ceremony Akiko was suppose to have a room of her own in the palace but as she doesn't feel safe except when she's with Zuko so she's moving into his room. And they already shared a room when they were in Ba Sing Sa. 

They just put one of those screens with a scene of phoenixes and dragons dancing in the corner so she can change behind it and they simply don't do anything but sleep side by side without dishonoring themselves. Just as they were allowed to sleep in the same room side by side when in Ba Sing Sa by Iroh.

They do that here but they show themselves respect and honor more for their true father Iroh and their mother Ursa sake then the living death that is Ozai and Akiko's spirit animal friends Ting Ting and Kai keep an eye out all nigh to make sure Death that is Ozai and the Demon that is Azula don't come anywhere near them during the night. 

So they are safe when they sleep.