I thought I would post-back what I did to resolve my problem (I hope.).
I have a 400 mile drive ahead of me before I can test the solution, so I do not know whether it will work or not. I bought a 5-port Linksys SD205 10/100 switch to put upstairs. I figure that the DGS-2208 downstairs isn't exactly broken--although it will not auto-negotiate over that flaky wire--so I am not going to change it. The Linksys SD205 is supposed to have some repeater-like features in it, which should be helpful. The SD205 did not receive consistently high reviews, but I could not find any 10/100 switch which did.
Thank you all for the help!