I found a better solution on
this thread: to hard reset older Kindles, you need to hold the button for 30s (the charging/power light will go out, keep holding), then release, then hold for 10s. Basically, it takes 2 or 3 long presses to reset, not 1x long press as suggested in many places. The screen will flash black, then the little boy under a tree will appear and it will reboot. Much faster than disassembling and jump-starting the battery.
This worked to get both blank-screened Kindle 4s back up and running.