Read http://www.techrepublic.com/blog/data-center/diagram-your-network-with-these-tools/
As to why the internet is slow it's usually something I look into installing a SQUID cache server. It's rare to see high enough speed internet for most locations so we augment it with SQUID. Your IT staff should know this (or tell them to get up to speed.)
Hi,
I’m a teacher and my school’s LAN has been set up at different times by rather sloppy people.
It doesn’t work properly (the internet is extremely slow at times) and I’m trying to figure out what is the problem.
For the time being, I’m trying to write down the LAN structure (cables, switches etc.)
I’ve made a diagram on a piece of paper and I’d like to know if there is any good software for making LAN diagrams.
Thanks in advance! 
Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic