I have LXDE, yes. For the moment - I put 10.6 on with LXDE so I could use the laptop while waiting for the adapter. When I know everything is feasible I'll wipe this.
It has arrived today and I'm sitting here in front of three computers, all with Debian. Somebody else's Cinnamon-equipped laptop, laptop number 3, works with this adapter, a dm9601 clone. So I just need to work out what's missing, or maybe find the name of the interface that comes about as a result of this adapter? I've forgotten what I knew three or four days ago... I will make a start on the googling oddyssey regarding this, but hopefully if I am bald in a few hours one of you will swing in, in loin cloth, and dangle some clues - I'm getting Tarzan and Poirot mixed up now but anyway - or simply slap me out of what will be my misery by then and provide a/ the solution. But I am happy to struggle for ninety minutes or so.
Ooh. I think I have a usb drive with Deb/Cinn on, that might be a good start, might discover at least the interface name, he said doubtfully. The dm9601 module is present though in what I have on the ethernet-free laptop.
Still, at least the adapter works, that's got to be about 1/20th of the battle.