|but when I have connected the other end of the cable (I have tried three different cables that work with my laptop) to my modem or to my router, I see no lights flashing for the corresponding port on the modem/router|
This points toward a NIC issue.
Check for updated drivers for the NIC. If there are some, apply them. If it works, great, if not, the next thing I would do is uninstall, then reinstall the TCP/IP stack on the NIC. If that doesn't work, you could try going in to Device Manager and unistalling the NIC itself. Then you would reboot and let windows find the NIC and install the drivers.
If none of that works, then you should buy a new NIC and install it - remember if you do to disable the onboard in the BIOS.
It matters not how straight the gate,
How charged with punishments the scroll,
I am the master of my fate;
I am the captain of my soul.