Needless to say, the removal of the Homegroup did nothing to help the wireless cause, and I knew that it wouldn’t. But one has to try these things anyway.
This morning, I used SIW to identify the wireless NIC. It is a Realtek 8191SE, and it just so happens that Realtek have a driver for the adapter on their website. It installed without issue, and the Toshiba connected to the wireless router immediately. Not sure if this is a good sign, but fingers crossed that it will work.
More to follow..