Where is the network traffic coming from?

Bad performance in the internal network could of course make internet feel slow. But it is more likely that the internet connection is overloaded. Especially if you don't experience any slowness in the internal network at the same time.
How you hunt down the bad traffic depends on which equipment you have.
1) You could install wireshark on a pc and mirror traffic in a switch to that pc.
2) If you have control of a router in the path you can have that router sending netflow messages to a netflow collector.
3) You firewall might have built in logs of the traffic flows.

We do have the netflow analyser but it only marks spikes in traffic as a whole, and I would like to track it down to IP address to find out who it causing so much traffic. We have a cisco router, but I would hesitate to mess around with it all. For that reason, I am just trying to find out how to get an IP address to see whos downloading masses of data.

Any decent network monitoring software will give you the answer.

PRTG: http://www.paessler.com/tools