802.11b compliant doesn't mean anything except that it is able to run at "B" wireless speeds with B wireless capable routers. You need to know the actual physical chipset inside the adapter (there are many different chipsets used by different adapter manufacturers) in order to pick a driver that will work with the adapter.
Windows does not have built-in drivers for most wireless adapters, and even if it has a driver from another manufacturer that will work with your adapter, you'll still need to know the chipset in the adapter in order to pick one that will work.
Does the adapter not have a brand name or model number on it anywhere? What about a bar code label with an FCC ID number? Even a serial number might help identify the manufacturer of the adapter, which could at least be a start in determining the drivers you need to use.