This spoiler free chapter of the guide to The Last of Us 2 has a walkthrough for Ellie's visit to the St. Mary's Hospital.
Going to the hospital
Keep swimming until you reach an area with a lowered tree branch - swim beneath it and continue into the wolf base. After the cut-scene, run up the stairs and you will see your opponents pack supplies into an ambulance.
Exit through the window and turn right. Then hide behind the tent, wait for the guard, and kill him while he is still behind the tent. Enter the tent and start eliminating targets in the area. There are plenty of enemies so be careful - try to clear the area around the hospital first and then go inside to kill more opponents. You can hide, wait for your enemies to start looking for you again and then go back to killing them stealthily.
Go further into the hospital and kill more enemies. Finally, go through the slightly open door. Upstairs, use the ventilation shaft to move on. Eventually, you will come across the exit of the shaft. Jump down, and after a while, you will find Nora.
After the cut-scene, run after the woman. Jump over obstacles and run after Nora. When she barricades the door, break the window next to her and continue the chase. Don't pay attention to the enemies shooting from above - keep running after your target.
When you finally catch up with Nora, you two will start fighting - keep pressing the Square button quickly to win. The woman starts running again - go after her. Eventually, you will both fall into a fungus-infested floor. Armed opponents will also fall down - attack them with a Molotov cocktail. This will very quickly lure the mutants to them. Wait for the outcome of the fight and kill the remaining opponents. Then get to the door and follow Nora.
After the cut-scenes, you will see another flashback with Ellie. Go to the hospital and search the rooms. Your goal is to get to the bag where you will find a recorder. Watch more cut-scenes.