Once you've made your way to the island collect the money located next to the fountain and then proceed towards the gate. Interact with a hook found above the gate to get to the other side and then enter the tower.
You'll notice an opened locker in the first room to the right of the entrance and you should definitely search it, because you can find Voxophone #13: Tiger by the Tail - Ty Bradley. Explore the rest of the room to find less important loot. The room with the Passive Siphon doesn't have anything interesting so you may head further into the building.
You'll end up in a corridor leading to three different rooms. The laboratory to your left contains Voxophone #14: To: R. Thompson Re: Fuses - Ty Bradley, as well as a bottle of salt and a medkit. When you're done searching proceed towards the proper Siphon.
There's a Voxophone #15: A Reward, Deferred - Ty Bradley on the desk in this room and you can also find Infusion #4 just behind it. Collect these items and then move towards the elevator located on the other side of the room.
The elevator will take you to Specimen Observation Room #1. You won't anything useful here so proceed to the corridor located to your left. You'll notice a Specimen Tracker on the wall and after using it you'll find out where to look for the girl. Step through a hatch to get to the next room.
Pull a lever after entering Specimen Observation Room #2 in order to see Elizabeth for the very first time. Head to the next hallway, use the Specimen Tracker again and then proceed to Specimen Observation Room #3. You'll see the girl here one more time and once that scene has ended resume your journey.
Specimen Observation Room #4 has yet another two-way mirror, however you'll also find a collectible here - Voxophone #16: The Source of Her Power - Rosalind Lutece. Take the item and proceed to the top of the tower.
Once you're outside use a small walkway leading to a room located inside the statue's head. You'll get to the Library from here and that's where you'll find Elizabeth.