Thank you for a quick reply.
As for the suggestions offered:

Yes, I did. And Windows sound recorder worked (no warnings; the "start record" button worked as expected) but didn't record the mic signal. As it turned out, somehow in Windows 8 it has a Level setting equal to 0 on a Levels Tab in the Properties window of the Record Devices system window for my chosen mic input Analog(1+2). It's a mystery for me, but whenever I pulled the BabyFace's USB cable off and then plugged it in again, or rebooted the system, the Level was reset to 0 again. Setting it manually to a higher value of the suggested range resulted in live input levels in Windows sound recorder and it did record the sound. But, alas, it wouldn't help with DNS.

As far as I understand, the problem with DNS resides in the fact that BabeFace doesn't register itself as a microphone in Windows. And while my DNS application can see this audio device, it can't for whatever reason consider it a valid input (a mic).

To make my case more interesting, I can tell you that I have DNS 12.5 working just fine under Windows 7 on another notebook with the same BabyFace and mic used. The difference is with Sound card and DNS version - in 12.5 version the layout of selectable input devices is totally different and I can select one of the several options. The non-working options are: Mic-In and USB Microphone. The working option turned out to be a Line-In (which is counterintuitive to me). But it's fine till it works. The built-in audio card is "IDT High Definition Audio" (as it's listed in Device manager).
My "problematic" notebook (ASUS N56V with Windows 8) has a notorious Realtek High Definition Audio and there's no such option as "Line-In" in "Manage Destination Sources" window of DNS. There is a "Stereo Mix: Realtek High Definition Audio"  and  "Microphone: Realtek High Definition Audio" (the latter works OK, by the way, not taking into account poor sound quality, so I have to turn off the built in Microphone in the Recording Devices Tab of the Sound window to try work with BabyFace).
Maybe this is somehow related to my problem? Any additional ideas or suggestions, please?

Never knew it existed, thanks for an opportunity to learn new stuff (I'm a newbie). But no, it was switched off.

How can I do that, please?

Please help.

This is only my guess, I don't know for sure. It seems after reading so many posts on forums here and there, that I've just read this opinion somewhere. It might as well be wrong.
But the problem is still there and I'd like to find a solution. The cheap mic that I have is working with the Realtek sound card, but it's useless because of the poor recognition quality. So, I hope that here – on forum where people are "more friendly" with sound engineering – I'd get an answer.