Hospital
The citadel was their best option to get help for Kaidan. He was in bad shape. Liara had done what she could to stabilize his condition but he needed care from professionals.
After meeting with the council, Shepard went to visit him. She found herself overwhelmed with emotion. Part of her was still angry at him for accusing her of working with Cerberus back on Mars. But this was also her friend and she wasn't about to watch him die.
So she told him to fight, because Shepard knew, deep down, he was a fighter. He'd pull through and then he could rejoin her on the Normandy, one last time.