I then had to tell both laptops to forget they had ever spoken to the wireless connection (because they were both failing to connect, and not allowing me to do anything useful) - which would have been easier on my laptop if Windows 8 hadn't removed the "Manage Wireless Connections" option, requiring me to download an app to do it.
And then I connected, and my speed has increased from around 20MBit to around 40MBit. Which still isn't earth-shattering speeds, but will make copying video files around less painful.
Cheers to various people for their suggestions!
Original post on Dreamwidth - there are comments there.