First off at the very least I'd get a fecal done before letting them share anything or interacting with each other.
You need to take the female out and put her somewhere else for a week or so. Put the male into the large cage by himself. He needs to claim it as his territory.
Then I would introduce the female into the cage. If you house a female alone and then put a male in her cage she will defend her territory which is bad. In the wild the males have territories and the females roam through but do not establish their own.
I would introduce them on the floor in the meantime that way there is plenty of room for her to run away if she finds the male unacceptable. He should bob his head and she should bow. Looks like she is doing pushups. If she wants him to mate her she will keep her back legs straight so her butt is in the air. If she is not ready to mate she will keep the back of her body against the ground.
They both may flatten and tilt their bodies toward each other and then circle. This is not ideal but may happen if the female is particularly dominant or aggressive. I don't let this go on very long. This behavior is where dragons get injured. This is likely what would happen if you put the male into a females cage.
Ideally she should be bowing and he will run up behind her and bite the back of her head or neck and flatten her to the ground. She will flail her arms like he is killing her but it's OK and don't stop them. Then he'll either mate her or let go and move away satified that she now knows who's boss. lol.