I measured it while running, both with and without the keyboard. I don't get what you mean by "turn the computer off, and measure it with a multimeter".
There's a switch in the back of the computer. Turn that to the *off* position. Unplug the keyboard. Also unplug the power cord from the system. Open the system. Locate the resistor. Switch the multimeter to "Ohm", written as a big omega. Connect one wire of the multimeter to one end of the resistor, connect the other wire to the other end. What do you read?
I should probably mention that (of course) there has been a battery leakage. Battery is removed. Have seen much worse damage than this though.
Then apparently there is more damage than you assume. Really. That's a number one reason why strange things happen. Just changing the resitor will not provide much. Keyboard power and wiring is pretty much in the same position as the real-time clock, namely in the front-right position of the board.
In that case, remove the board completely and clean carefully, inspect all wires. It wouldn't surprise me the least if some of the connections seem to look good at first inspection, but fail to connect properly because they were corroded by the battery.