I've not played White but if it works in the way DSi connections usually work then you need to make sure your XL already is set up to that router by going into:
->Settings
->Internet
->Connection Settings
-> Advanced setup *the important part*
-> set up the connection on slot 4, 5 or 6
Once the XL is set up and can connect on its own then the game should be able to connect through that connection when you try to use a Wi-Fi function.
If it's something different then I'm afraid I don't know.