We know that nobody can enter or exit the birdcage without being cut, so how did Jora get inside the birdcage?
Giolla was in the birdcage since its erection.
In episode 656 of One Piece, Trafalgar Law has Giolla as a hostage.
After Law realized that Doffy would have sacrificed Giolla if necessary and thus holding her hostage held no benefit, he decided to release her. She tried to stay but Doffy told her to leave so she ran back to Dressrosa.
The birdcage was not surrounding Dressrosa until episode 680.