The pins are exactly the right pitch and locations, but you also need to put resistors on half of them, so I cut those pins a little shorter and added SMD resistors, which happen to be exactly as long as the pins.
I also had to unsolder all the LEDs from the board itself.
Even though I removed the LED from pin 15, the corresponding column is still a little bit dimmer than the others, I wonder if there is something I can do about it. (Turns out it was a bad soldering joint.)