Without the use of any software, what would be the cheapest way to connect my broken laptop's keyboard to my desktop?
This is like asking how you would change the electrically wiring on an HDMI cable so it plugs into an PAL/NTSC television. In other words, it assumes everything is the same except the wires are swapped around. Obviously this is not the case. If you are really lucky the USB in your laptop keyboard is self-contained then you just run those signals out to a plug, but then it is just a keyboard like any other that you can buy for $20. Not some special rebindable one.

Besides, I think laptops almost always, if not always, have the keyboard USB controller on the motherboard such that the keyboard itself is just a dumb matrix keypad.

What you are asking for is a custom "laptop keyboard USB controller" which is not something that can be built without a firm knowledge of hardware and software. Buy one or build one from an open source project. Just make sure you find one that allows rebinding in the controller itself.

Frank Adam's Project:

https://www.youtube.com/watch?v=Z1PheqSNNP8

https://www.hackster.io/frank-adams/laptop-touchpad-conversion-to-usb-d70519

https://www.instructables.com/id/How-to-Make-a-USB-Laptop-Keyboard-Controller/

https://github.com/thedalles77/USB_Laptop_Keyboard_Controller