I looked up the datasheet for the Li1820 which is, apparently, virtually identical to the Li1818 and discovered that this uses a different wireless card, the Intel© PRO/Wireless 3945ABG. This card is installed in numerous other laptops of various brands.
I borrowed one from an Amilo Pi1556 (which also has an external wireless on/off switch), installed it in the Li1818 and it was recognised and installed in Windows 7 automatically - no searching for drivers required. It works perfectly, just like in any other normal laptop

A quick search on eBay showed dozens of these used wireless cards for sale for very little money.
Hope this helps someone.