so you're the ONE still using dial up.
just kidding. the lan adaptor won't work because it's fitted for an ethernet cable. Are you talking about the usb dongle? that may work but i've heard some bad things about them anyway. You're best bet would probably be to pick up a wireless router and connect through that. Kind of pricy though spending at least $50 to get the wii online. maybe just try the dongle and see how it does, then if it does work you've only spent $20, and if it doesn't you might have a chance of returning it.
hope that helps.
p.s.- i don't have experience with dial-up so i'm not 100% sure if the dongle will work, but i don't see why it wouldn't it just takes the connection from your computer and transmits it as a wi-fi signal, it doesn't know what's plugged into the modem though.