I'm at work so I can't check it right now, but I know that both computers have connections set to "home network" or whatever the right term is, not public place or the other options.
On the desktop, the connection is listed as "Local Area Connection". On the laptop, the connection is listed as NETGEAR. This is the name I gave when I originally set up the router so that the laptop could use internet. I assume the settings for that reside in the router since I've never had to set it up again since the first time.

Are you saying that if I set up the homegroup on the desktop, wait for the password, change it to what I want, then do the same thing on the laptop (using the same password) that they will in fact be in a homegroup with sharing enabled?