What you are trying to accomplish with the Edimax is what is called a client bridge. Unfortunately most home routers are designed (in software) to be the Access Point of a structured network, not the Client (which is the other end - think jack and plug). Loading aftermarket firmware, like OpenWRT or DD-WRT will allow you to put the units into Client mode, but unfortunately it doesn't look like either of those routers is supported as far as I can tell with a quick look.
For a relatively inexpensive alternative, check one of these
USB attached client interfaces - I'd personally suggest one of the WiFiStation units listed near the bottom of the list if the range is long or there are more than a couple of walls between the router and your computer. For Ethernet connected options look for devices that are advertised to connect game consoles - those are client bridges, but unfortunately the vendors are very "proud" of their work and sell them at a real premium. Otherwise find yourself a Linksys WRT54GL on sale somewhere and install OpenWRT or DD_WRT and make yourself a client box, but expect to spend a little time learning to figure it out if you are not at least a pretty involved hobbyist. (If you don't know what Flashing Microcode with TFTP means, you will have some learning to do.)