This link does not reveal anything about the hardware / drivers involved.
Once the drivers are installed, at least one out of the four channels should show up.

Check if you can find your card in the Device List

Message has been edited, click here to see original content

I tried. Really tried. But after reading that eBay advert twice and staring at the photos I can not see any Manufacturer or Model Number information there. Without that info I can not assist you because I have no clue what that card really consists of at any meaningful level.

I find it rather odd that nowhere does that eBay advert contain the words "WDM Compliant", nor is there any mention of DirectX / DirectShow. Leads me to think the card may be a "bastard" design that's not really compatible with anything but its own software.

Try running wmcap.exe (it's part of the standard Windows Media Encoder distribution) and see if it can recognize your card. If the card is compatible and the driver is correctly installed, wmcap.exe *will* see it.

Edited to add:

If your card is one of the many "Brand X" products that are based on Conexant BT878/9 chip, you probably need this to make it WDM capable:
LINK TO DRIVER AT SOURCEFORGE

Note that LogiSphere supports hundreds of WDM compliant devices, but there are also hundreds of non-WDM compliant Brand-X devices out there being sold cheap at places like eBay (because their original owners found out they don't work with much of anything).

If you can determine what chip the card is built around (BT878 is a good possibility) there still a chance you can get it to work, provided you are willing to put some time and effort into it.

There are open-source drivers out there to take care of all sorts of compatibility problems. Google is your friend! What you seek is a "WDM Compliant" driver for your card's chipset. It probably exists somewhere.