I have an older xbox 360 and the ether net port on the back will not read if there is an ether net cable plugged into it. and
Your best bet is to see if you can buy a cheap refurbished one at this point and replace it, MS wants $100 to repair it. you might find a third party to do it cheaper, maybe even places like gamestop will help you, since they basically repair and refurbish them then resell them.
The other option is buy a wifi dongle and use that, it plugs into a USB port, but again costs about $100 every time I see them, though I have seen a few used in game stop for about $50, still not a worth while investment at this time as the 360 is soon to be replaced with the xbox one.