My intent on using a cable was to help determine whether there is a working Internet connection. The other computer works with the cable so there is certainly Internet connectivity and the cable is not needed. If the laptop is able to connect using a cable, we still have to get the wi-fi fixed.
We've now determined that you are recieving an IP address from the router, but no Internet connectivity..
Restart the computer and tap F8 until you see "advanced boot options", select "safe mode with networking" and press enter.. When it boots up, try connecting to the Internet with wi-fi..