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Once you wait through all of the cut scenes, approach the moving lion statue. Two ghosts will make tracks into the building, shoot at them, and they will flee again. (Insert a cowardly lion joke here). Give each one a try and then equip your PKE meter. Head down into the lobby and take the door on the right. You are about to enter the Reading Room.
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Book Golem 1
Once you get to the front third of the room, your book golem will appear. This one is stronger than the ones you have faced in your previous golem encounters. That being said, your strategy remains the same. Wear him down (your new attacks are handy) and use the capture stream to slam at his focal point. You will encounter a few other golems in this level, and all will be taken down the same way. Once this is done you may have the urge to bail. Before you do so open the door on the far left hand corner and PKE it to get a possessed portrait.
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Chasing The Grey Lady
You will follow the Gray Lady through a few different areas of the Library. When most of the time, you might have to listen to the original Ghostbusters whine about how she keeps shushing you. You will be getting near the end of this chase when you reach a room where the bookcases start moving to block their path. Just keep walking the maze, and taking the open option until you reach the other side of the room. Yes, this means that you may end up back tracking every once and a while. No, you cannot use your proton pack to speed things up.
Eventually, you'll find yourself confronted with The Gray Lady once again. In a very frustrating move, she once again leads you deeper into the library. Continue with this game of chase for several more rooms. Don't be too thrown off when she summons both book bats and golems to make your life difficult. Handle this the way you would any other proton pack fodder and keep on chasing. Make your way into the children's reading room and use your PKE meter to find a trophy, there will also be one in the special collections room. Eventually, you will be confronted by two unpleasant ghosts with shields. Capture the shields, and toss them away before you attack them. Otherwise, you won't do any damage. Once the room is clean, put your meter back on and use it to find the secret passage in the back.
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The Secret Passage
You will head your way through the first passage and enter a room. When you get there the Gray Lady is gracious enough to open a second one for you. Head down that passage, and when Egon gives you the go ahead, rip off the boards. After you blast your way through several more golems, you will be given some new equipment. It's a kind of anti-slime slime gun. Use it to a neutralize any of the toxic black slime you encounter, and move forward. Once you wait through the cut scenes, it will be time to defeat The Gray Lady herself. This battle will be surprisingly easy. That is when your attention will be drawn to a glowing orb portal, which is going to take you to a parallel dimension New York Library.
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The Other Side of The Looking Glass
You will see a set of stairs right in front of you. Take them down, and you'll encounter a golem. You have to fight them alone because no matter what you do, you're going to lose contact with the other guys. As you learn this is going to be a trial and error game of 'find the right portal' to get home. No point wasting time, so get walking. Head to the far right side of the room, and after defeating the book golem that will spawn, head through the portal. Start heading straight forward and eventually you'll come to the bridge. When you get to the bridge you'll learn how to use a new toy. This is great if you enjoy puzzles, because you are about to solve one. Just kidding, it's not that hard. Your new toy to pull the bridge down so you can proceed. Keep heading forward until you reach a circular room with a large wall.
Use your goo tethers to enter this room and reveal a new portal. Enter this portal, and take the corridor that bends to the left. When one of your compatriots heads into the portal in front of you there will be a little short out and you find yourself in a hall of mirrors. A particularly annoying enemy will use these mirrors to ambush you. No point in making his job easy, so destroy all of the mirrors, then trap him. From this point on, there is only one path you can take. No matter where you go you'll end up heading to the same location. You will know you're getting close when the other Ghostbusters are telling you that they found their way out. Keep going forward and you'll see the long staircase. Getting up it won't be as easy as it first seems. Bats will attack you at every possible turn, and no matter how many you kill they will just keep re-spawning. It is a timing puzzle, one where you have to get through the spaces before the second wave comes. Once you get to the top, use the tethers to bring the portal in range.
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You find yourself in a pseudo-normal library. You are going to have to defeat the collector, who Ray manages to inadvertently provoke into showing itself. Apparently, this man does not know when it is a really good idea to shut his mouth. Once the clutter makes his appearance you should settle in for a long fight. Start by using your slime pump to rid him of a protective coating of black slime, then back up quickly. Your job is to use your capture beam to grab the mask off his face, and break it. Just do yourself a favor, and don't try to remove it until your compatriots have his hands restrained. If you go in too early he will just put it right back on. After you take care of these two layers your job is mostly the same as it is in any other boss battle.