Ed, I'm running WPA2-PSK. Not tryed it on a open network..I doesn't matter if Its plugged into the mains or not, problem still there Jason..
So according to the model number you shouls has this wireless adapter fitted to your laptop, on most laptops they are easy to replace, look under the laptop you should see a few covers, they are generally under there or on some models they are under the keyboard.So this should be your current cardhttp://www.ebay.co.uk/itm/SONY-VAIO-VPCEE3E0E-Wireless-Wifi-p-n-T77H126-06-LF-/220776121963?pt=UK_Computing_Laptops_EH&hash=item336748466bAll you do is remove that and replace with the same card design, best try to keep to the original laptop manufacturer, found this which is Intel brand but also used in Sony'shttp://www.ebay.co.uk/itm/Sony-Genuine-VGN-CS-Wireless-Card-Intel-512AN-HMW-/280701393000?pt=UK_Computing_LaptopAccessories_PowerSupplies&hash=item415b1b5868When buying replacements cards make sure that they have the same amount of pins and that the size of the card itself is not too big for the slot and for the space available.That should sort you out if the wireless card is faulty. Let me know if you need more help fitting, one tip is to remember which cable goes were, take a pic or write it donw before starting.EDIT: Also make sure the new card is compatible with your OS.
With what encription? AES or TKIP?