You may wish to try this first. Type "run" ( not in quotes) in the Search Box
(make sure that you have your modem/router is on and connected to your laptop )
When command window opens type ping 127.0.0.1 then press enter. Check result.
Do the following and check results
Pinging 127.0.0.1 (loopback) tests that your TCP/IP stack isn't corrupt.
Pinging your ip address (e.g., 192.168.1.100) tests if your NIC is functioning.-
(ip address can be found by entering ipconfig /all command - usually listed as IPv4 address)
Pinging your gateway tests if your LAN is working.
(gateway address can be found by entering ipconfig /all command - usually listed as default gateway address)
Pinging an external address tests if routing and your internet connection are working.
e.g. type ping google.com
If no results then check in Device Manager that the Network Adapter is working properly,
i.e. no red crosses or yellow exclamation marks.
I do not know if in a laptop the NIC is a separate card, most likely it is intergrated into the motherboard and therefore if faulty would probably require a replacement motherboard. One thought it that perhaps because, being a laptop, the network cable is being removed more often, the RJ45 socket connection on the motherboard may have become loose. You say that it lights up when you first plug in. Maybe you should try holding it in, perhaps with a little pressure in various directions and see if it maintains a connection. If so it would certainly point to a faulty connector which should be cheaper to fix than replacing a motherboard.
Also would using a USB modem or WiFi be a viable alternative for your internet connection method?