My wife and I both have one for our laptops. There is a small dongle which goes into any USB socket on your computer; the driver will install automatically and you're in business.
It is a wireless mouse, current keyboard is a Microsoft model but it is not functioning with the new mouse. I never thought that mouse and keyboard would come as a pair.
I have plugged the dongle into the PC. The mouse scrolls ok but the keyboard does not respond i.e. no characters appear. The mouse will not delete e-mails either.
One can have a situation where both the keyboard and the mouse can connect to the same dongle. I have a Logitech solar keyboard and Logitech trackball and both came with their own dongle. But it said I could run off the one so that's what I do. It's the dongle that can cope with multiple wireless connections regardless which they are.
If your keyboard is wireless as well as the mouse and they are different makes wouldn't you need two dongles! You have said there is only one dongle connected!