Well it turns out my dongle can be recognized if I bend it really hard while in the USB port (and it does send notification when that happens). Not sure if I somehow damaged it mechanically or it is this way because of two years in the closet.
Ann, the keyboard behaves the same for me, green light for some time after putting in the batteries. Only thing I can recommend is to install the microsoft drivers for your set (https://www.microsoft.com/accessories/en-gb/d/wireless-desktop-2000 for Wireless2000).
Too bad the dongle is not replaceable, I'll probably try changing and re-soldering the USB plug and then I'll probably end up ditching on microsoft hardware for some time...