|Having two routers on the same network is possible but very complicated and beyond my knowledge.|
A much easier solution would be to log in to your existing router's configuration web-page and, if available, change it's wireless encryption type from WPA2 to WEP.
Most routers (including my brand new one) have WEP as an alternative to WPA/WPA2.
I don't know the make of your existing router (the one your currently using) but with most routers you access it's configuration page by typing http://192.168.0.1 into the address bar of any web-browser, then press Enter.
If that address doesn't work, check the router's user guide as it may be a different web address.