Ok, finally got it operational. (That was easy!

) Thanks for your help Aldanga.
To summarize for people coming from a Google search:
Issue: You want to share media from a desktop PC to an Xbox 360, but the 360 is too far away from your router and does not have wifi. However, you have a laptop with wifi and an ethernet cable to connect the two.
Solution: You must create a
Network Bridge from your laptop on your ethernet and wireless connections. In my case, once I did this, my laptop completely lost connection (no IP address). You can fix it using
this support article. Once this was working, I had to change my 360 from Automatic to Manual network settings, and assign it an IP address outside of my router's DHCP range. (For example, if the range is 192.168.1.1 - 192.168.1.150, you could use 192.168.1.200). Be sure to also assign the Primary DNS Server. (Your router's IP.) Once all this was in place, the 360 showed up as a device in my network and I was able to add it as an Extender.
Special note:
Internet Connection Sharing will not work because it puts the 360 on a different subnet than your router. Unless you do some magic in your router settings that enables cross-subnet communication (which my router did not support), I don't think it's possible to get this method working.