To get to the hospital, first you need to reach the train station. Head out from the ferry in the direction of Steiner's workshop, turn left once you reach the workshop.
Make your way to the stairs, but before you go up, head left, into the back alley - behind an opened gate you'll find a wooden cart. Help yourself to the pair of wedges that block its wheels (just pick up one of them, the other one will find its way to Kate's inventory by itself). Once the cart rolls down, pick up one more wedge - the small one - from where it was standing.
Turn back and go upstairs (if you head to the other side of the square near the station and go downstairs, by the city hall, you'll have another chance to talk to the mayor, assuming you've parted on good terms- Brave... in good time! achievement).
When on the square, head for the building on the left. Go across the fence and approach the building. Try to open the door. Unfortunately, it will be locked.
Forcing open the station building door
Look closely at the crevice under the door and slide in the wedges, first the small one to the left, then a normal one right beside it (a new active point will appear). Now extract the small wedge and slide it on the right side of the door, then another "large" one (again, a new active point will appear on the right). This will open the door.
Enter the building. Before you put your hand on the panel right of the door, focus on the power box hanging on the wall (left of the door). Zoom in and try to open it - to no effect.
Take your knife and slide the blade into the crevice on the right (upper active point) and use it to pry open the box cover (move the mouse down). Move down the red slide (move the mouse) you'll find inside the box.
Once the control panel near the window lights up, zoom in and move down the switch on the right. A train will come and stop right by the window. Exit the building, head right and enter the train car through the open door on the right.
Once inside, go left and approach the driver panel (left of the large metal wheel; click on the active point).