Well, it could work with just a system of cords and pulleys..
Have a huge, central library-information storage area, with miles of cords extending outwards to various out-posts.
These out-posts are small buildings with several technicians, and a secretary behind a desk. You ask him the question. If it's a common one, he'll just pull out a few scrolls on hand and tell you what the answer is. If it's something more obscure, then he'll send the question onto the central library.
Which he does prodding one of the technicians in the Lever and Pulley Room. Each of the various levers and pulleys sends a different number and set of pulls along a cord that goes to the central library. The trained technicians will then send the question to the next outpost closer to the central library, which will relay it on.
Of course, once it reaches the central library, a system of pulleys will operate to clack two blocks of wood together. A technician will listen and write down the pattern, which a scribe will translate into actual letters and words. Then someone goes off to find the information.
Then they do the whole thing over again, in reverse.
A rather inefficient system, but I'm sure the Durhians will improve upon it.
“She'd become a governess. It was one of the few jobs a known lady could do. And she'd taken to it well. She'd sworn that if she did indeed ever find herself dancing on rooftops with chimney sweeps she'd beat herself to death with her own umbrella.”
― Hogfather