Nightwing could also be a likely savior. He was once Robin himself and has a strong bond with the current Robin. He might take it upon himself to rescue Robin, using his own unique fighting style and experience to outwit the Joker. Plus, he has his own network of contacts in Gotham that could help in the search.
In one scenario, the Joker might kidnap Robin to taunt Batman. He could hold Robin in a secret, elaborately booby - trapped lair. Robin, being the resourceful sidekick, would try to find ways to escape and leave clues for Batman. The Joker would then play mind games with both Robin and Batman through recorded messages and false leads.
One idea could be that the Joker kidnaps Batman to force him into a series of cruel 'games'. Maybe he wants Batman to choose between saving his closest allies or innocent civilians, testing his moral limits.
One possible fanfiction plot could be that Deathstroke kidnaps Robin to use him as bait for Batman. He wants to lure Batman into a trap and finally defeat him. Robin, being the resourceful sidekick, leaves behind small clues for Batman to find him, adding an element of mystery and suspense to the story.
Robin would probably be both scared and angry. He's a brave kid, but facing the Joker alone is no small feat. He might initially try to fight back if he has the chance.
Perhaps the girl outsmarts the Joker on her own. She uses her wits to escape from the place where she was held captive and then leads the police to the Joker's hideout. So, the fanfiction ends with the girl being a hero herself.
One possible plot could be that the Joker kidnaps the girl to use her as a bargaining chip against Batman. He might think that Batman has a soft spot for this girl and would do anything to save her. So, he holds her in a secret location filled with his signature traps and chaos.
The plot might go like this. Slade kidnaps Robin in a very stealthy way. He takes Robin to an abandoned building. There, he tries to indoctrinate Robin into his way of thinking. He could be telling Robin that Batman is using him and that he can offer Robin more power. However, Robin stays true to his beliefs. Meanwhile, Batman is on a frantic search for Robin, following every lead, and the Bat - family is also on high alert. Robin is constantly looking for an opportunity to turn the tables on Slade and escape.
The Joker might try to make Batman laugh at his own jokes as a form of humiliation. He could use his trademark laughing gas in a way that forces Batman to either laugh or suffocate.