Is the LAN port on the motherboard or an adapter? If it is an adapter, try reseating in, make sure it fits securely and level with the motherboard. You can try pinging the adapter in a command prompt with "Ping 127.0.0.1" if you get a time out then the adapter is probably bad.
You can try updating the drivers, you will have to download the drivers from the manufacturer on another computer to a USB drive.
If it the LAN port is on the motherboard, check the BIOS settings for the Ethernet adapter. If the LAN port is on an adapter, then try deleting the network device in Device Manager, remove the driver and then scan for hardware changes.
If none of these steps work, you may be able to get another Ethernet adapter. Check to see if you have an open slot, determine what kind of adapter slot it is (usually it will be PCI). Ethernet adapters are fairly cheap.