How do the prince and Cinder meet again? Why does Cinder turn down Kai's offer?

In the very first chapter of the book Cinder, our main character Linh Cinder meets Prince Kai for the first time when he brings an android in need of repair to her mechanic booth. Their encounter is very brief, but after Kai makes plans to pick up his android later on and leaves the market, Cinder catches herself thinking about the possibility of having a crush on the Prince.
Cinder is not your typical adapted Cinderella story. In this story, Cinder is a cyborg living in a world ravaged by a deadly plague called letumosis. While there is still the character background of being an orphan with an evil stepmother and she is treated cruelly by most of her step-family, Cinder's story becomes much more than being forced to work for her family. The government of New Beijing does not consider cyborgs to truly be people, so cyborgs are drafted to become research subjects for letumosis, meaning that those drafted will surely die from the disease. When Cinder's stepsister Peony is infected with the plague, Adri angrily signs Cinder up to be a research subject. Fortunately, it turns out that Cinder is somehow immune to letumosis, so becoming a test subject will not kill her.
During her time as a test subject for Dr. Erland, Cinder meets Kai again multiple times. In chapter 14, Kai comes in just as Cinder is passing out from the pain of one of Dr. Erland's tests. After she regains consciousness and Kai helps her up, they end up talking more about his broken android. In chapter 18, the two run into each other in the palace as Cinder is on her way to see Dr. Erland. After meeting with the doctor, Kai takes a moment to step to the side and ask Cinder if she would join him at the annual ball coming up in a few days. She turns down his offer for two reasons: the first reason is that Cinder and her android Iko are planning to run away from New Beijing on the night of the ball "if the car [is] fixed in time." The second reason is that up until this point, Cinder has kept the fact that she is a cyborg hidden from Kai, as she believes that he would think differently of her if he knew. Cinder knows that no one would approve of a Prince and a cyborg being together, and she thinks that Kai would be "mortified" if anyone found out.