Alex ends up at the library because he remembers Sayle mentioning that Ian "spent half of his time" (11.5) there.
He asks the librarian about Ian. The man remembers Ian researching viruses a lot, and Alex assumes that he means computer viruses.
The librarian also mentions that Ian spent a lot of time in section CL. Something clicks in Alex's head and he realizes that he saw the letters CL on Ian's map—Ian wrote down library reference numbers. Eureka.
Alex finds a book about the local Dozmary mine, which contains an old black-and-white map that fits the one that Ian had made. The puzzle pieces are falling together.
A photocopy of the map in hand, Alex heads to the entrance of the mine. Time is running out.
He finds a trapdoor leading into the mine and slowly climbs down the ladder; he reaches the bottom of the dark shaft and tries to "fight off the sense of claustrophobia" (11.45).
Alex follows Ian's map into the depths of the mine. As he gets deeper, the passageways become increasingly cramped and Alex can feel water at his feet.
It looks like he's reached a dead end until Alex realizes he's standing in front of a pool of water. Luckily, Ian left a dry wetsuit and a track of rope behind, presumably for his return voyage.
Alex suits up and dives in; the water is freezing and he can barely see a thing. Finally, he feels "his shoulder come clear and [...] he knew that he made it" (11.65).