|You appear to be using one of the desktop as an Internet host. That is the wrong way to do it. Fine if you only had two or three computers but cumbersome with seven.|
The correct set up should be Modem > Router > Switch > Computers.
Most home routers come with four Ethernet ports. As you have seven computers you need the switch to add the other three, There is no direct connection between the Modem and any of the computers.
It is the router that does the Internet sharing. It takes the Internet connection form the Modem and shares it out to all the computer via Network Address Translation.
Once you have the wired networks setup you can then connect the wireless computer. You need to read the instructions for the router very carefully. It is the router that does most of the work.