Go to the Device Manager function in the Control Panel of
your Start's main menu and click on the Network Adapter sub-menu arrow and, you should have one called rather Marvell Yukon, Realtek Pci-e controller or, an Intel/AMD LAN internet chipset/card for your RJ-45 LAN port. Double click on that and check how old is your Device Driver and, if older than 5 years, it would explain as why it can no longer connect to the internet. Plus, make sure a recent copy of a Firewall and Anti-Virus program is, also, installed onto that aging single core machine.
If you still are on dial-up/Dsl internal card modem, you do not have the use of the Network Adapter. Click on the Modem one, then on the name of such modem unit's name twice for bringing the window for using the Driver Tab. The Driver Tab will give you all of the information that you need for the current status of such device.
Now, knowing that you are constantly broke, use that other computer to download the AVG Anti-Virus 2012's Free Edition and make a DVD or CD copy of it. Install it on your single core machine. As for the Firewall, activate the already existing one called Windows Firewall in the Control Panel window. Once you get back onto the internet, go onto Microsoft.com's web site and make a search for the Free feature that came with Vista(5 to 6 years ago) called Windows Defender and download and install it. Defender will be your totally free app. for anti-malware/anti-spyware.
Giving me the little bit extra information on your current status, really helped me discern the real problem and, that way, I could come up with a real, productive, helping, solution. I am, myself, not that rich, either. Really do hope that helps a lot more than my previous posts. 
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