Is there a foolproof way to detect if my sound card supports 5.1 surround sound?

The 5 audio jacks you quote is the important thing, not the number of speakers.

It refers to the audio jacks on your motherboard or sound card, not the plugs from your speakers.

If you have multiple audio output jacks on your motherboard/sound card then you have support for surround sound (providing the correct drivers are installed)

The standard jacks are as follows:

Green - Front L/R
Black - Rear L/R
Grey - Side L/R
Orange/Yellow - Sub & Centre
Blue - stereo line in
Pink - mono mic in

2.0 will have 3 jacks - Green, Blue, Pink

5.1 will have 5 jacks - Green, Black, Orange/Yellow, Blue, Pink

7.1 will have 6 jacks - Green, Black, Grey, Orange/Yellow, Blue, Pink

In the pic below, clockwise from top left: 7.1, 7.1, 5.1, 2.0, 2.0

If you want to see in your system properties you need to open up the RealTek HD Audio Manager (from Control Panel or System Tray icon) and this should show an image on the right that will show you which are available on your system (jacks which are plugged in will be highlighted):

(Error here is because I am accessing this PC via RDP with audio disabled)