I removed the keyboard of my laptop (eMachines e725-4520) in hopes of fixing 12 dysfunctional keys. I followed the instructions of a couple of videos I found on YouTube. I've replaced keyboards of other laptops over the years.

I cleaned it gently with a soft cloth and a little 91% isopropyl alcohol. When I re-installed it it doesn't work at all. I suspected I hadn't seated the connector correctly so I re-tried that 10 times: it makes a solid mechanical connection.

How do I know whether I have ruined my keyboard or failed to seat the connector properly or made some other mistake or damage?

I find no report of failure in the log files. The /sys files for the 8042 keyboard exist, but they could just be reporting on the keyboard controller hardware.

One thing that springs to mind is there's usually a bios option 'Press F1 for any error'.
If you boot to the bios, and can't navigate, you have a keyboard disaster. You can plug in an external one and move around, it's the keyboard.
If you plug in the external and can not move in the bios, it's likely to be further in.
If it's the connector, I'll blame you:-). You should spot that.

I would set the F1 thing, reboot, and pull the external keyboard. What does POST say?

I removed the internal keyboard and can't find any difference in any log file or dmesg. The kernel creates the same keyboard i/o files.

The lesson from this is that internal keyboards are read-only devices: the computer doesn't write to them. This is different from external keyboards, such as all desktop computers have. I think that this is a mistake. It makes the connection difficult: internal keyboards have fragile multi-wire connectors; a keyboard with a built-in decoder chip would need only 4 more-robust wires and could communicate its state. The mentality of the laptop manufacturer is that you should replace your computer if the keyboard fails, I think.

Keyboards are run off a matrix of switches Rows x columns

So Row 1 column 2 is the key F1 or somesuch. That goes into a circuit which the keyboard sees as the keyboard. So your keyboard circuit might well be on the m/b.

Forget everything I suggested - it's the keyboard. Which bit of the keyboard(switches, plugs, wires or circuit), only substitution will tell you.