From what i know there are only 2 ways to get your Xbox 360 online.
1 Is a Ethernet cabel
2 Is the Microsoft USB Wireless Dongle.
You can only connect to the internet with the USB port by using the Microsoft USB Wireless Dongle, and since you say that your Ethernet port is broken i recommend you to try the following things.
1 Try the Ethernet cabel to your PC if your PC does connect via the Cabel
- Blow out the possible dust out your Xbox 360 Ethernet port.
- Check your router Firewall settings.
- Check if your Xbox 360 network settings are correct (IP adress etc)
2 If it does not connect to your PC
-Try a other Ethernet cabel.
I would like to know
-How long do you got this problem
-Did you changed anything on your Router or Xbox?
-If your Ethernet port would be broken (Which looks very unlikely) please explain how you came to that conclusion, did you do something to it?
Edit : I just found this topic which could help you http://www.techsupportforum.com/f142/xbox360-and-non-xbox-wireless-adapter-360951.html