One of my three fabulous sis-in-laws was in town and brought her iPad 4, which hadn’t booted up in over a month. When plugging the Lightning USB cable into the iPad and a power supply, the red battery icon periodically flashed on the screen, and when the cable was removed the icon also flashed that it needed power. However, no matter how long the iPad was plugged in for, it never stored enough of a charge to fully power up, even after putting it in DFU mode and restoring iOS to the latest version.

After reading this site and skipping about a quarter of the way down, the following bullet was helpful:

I held the power and home buttons for at least 10 minutes while the iPad remained plugged into the power supply and USB cable. Sure enough, about 30 minutes later the iPad booted up and continued to charge! Success.

Extra bits: why does this work? In the first paragraph of this post, we mentioned that the iPad kept flashing that it needed power, it was caught in a loop of sorts. By holding the home button and power button simultaneously, we are breaking this loop and allowing the iPad to continue charging. Woohoo!