I was able to achieve my goal of printing to my Brother printer wirelessly.
Just in case it helps someone in the future, I'm going to describe what I did.
I purchased a router that could be flashed with DD-WRT. I got a new TP-Link wireless N router on the cheap, Model TL-WR841N. (The difference between the WR841N and WR841ND is that the "ND" model has detachable antennas)
I'd never flashed DD-WRT before. It was quite a learning experience. Fortunately, the router that was hardware revision 7.2 rather than the latest one, revision 8. Apparently, DD-WRT has yet to be verified to work with the latest hardware revision 8, so I'm very lucky that I got sent an "older" router (which may have something to do with why it was sold cheap).
You really have to do your homework. I figured out that I needed to set up DD-WRT in "bridge" mode...but I found some posts on the DD-WRT forum from individuals who said they had difficulty getting bridge mode to work in the release of DD-WRT that was recommended for my router (model and hardware rev). Fortunately, they provided information on another release that worked.
(For anyone who needs the files I used: ftp://dd-wrt.com/others/eko/BrainSlayer-V24-preSP2/2011/12-20-11-r18024/tplink_tl-wr841ndv7/
I was able to find some YouTube videos that described how to set up DD-WRT in "bridge" mode, and these helped me get it done.
Here's the tutorial I used on how to set up bridge mode: http://www.youtube.com/watch?v=Jf140UmU39o&feature=related
I flashed the router using my laptop, and then, using the tutorial, was able to create the bridge, testing it by using the TP-link router as a "bridge" between my primary router and the laptop. (I.e., I turned of the wireless capability of my laptop and was able to access the internet successfully through my main router using the TP-Link as a bridge.)
After that success, I disconnected the router from my laptop and connected it to my Brother Printer. (And re-enabled wireless on my laptop.)
Finally, installed the LAN "version" of the Brother MFC-8480DN drivers on my laptop. During the installation the software scanned my network and--as a shock to me--actually found the printer, and configured it correctly. After installing the Brother software, I was able to print to the printer without any problem.