First, try to remember if there is anything that you have changed in your networking setup recently, even if its something minor.....
Well, let's try to work from the router back to the laptop....
You've said you switched ethernet cables (I assume the one from the EMachines to the Inspiron and vice versa?) If this still didn't work, try taking the ethernet cable from the Em and hook it up to the Inspiron, but plug the other end into a different port on the router, or at least the port the Em was connected to.
If that still shows no activity, check to see if your router has DHCP enabled for your laptops. If it does, check to make sure that DHCP is enabled for your Inspiron's NIC.
Pull up a command prompt, and type
See what the IP address is. If you've got your NIC set to DHCP, and it can get an IP address from the router, then it's probably something like 192.168.x.x
If it's set to DHCP, and can't talk to the router, then it will be something like 169.254.x.x
Well, if this is the case, then open the command prompt. Try pinging 127.0.0.1
If this comes back, then Windows can communicate with the NIC. Now, try pinging that address listed in the ipconfig....does that work? If so, then the network stack (what enables networking on Windows to work) is functioning.
*Remember, it sounds stupid, but open My Network Places, and make sure your Local Area Connection for your NIC isn't grayed-out. If it is, you just need to right-click and select enable.
Those are pretty hap-hazard instructions, but I'm needing a little more info to help you out. Post back with any info you get from that