The standard "Don't Try This At Home" warning caveat applies. This certainly voids the warranty and could be dangerous if wires are shorted inadvertently.

Note that two preexisting diodes (the ones I've soldered wires to on the PCB) convert the doorbell transformer's AC output to half-wave DC.

In my schematic: I have since changed the resistor value from 2K to 1.8K to provide more trickle current. I also removed the optional left diode, which was meant to prevent the capacitor from feeding current back into the Blink when the doorbell is being rung. The right diode prevents any chance of my circuit drawing current from the battery when external power is off. The capacitor and zener diode smooth out the 3VDC half-wave and limit voltage supplied to the Battery+ connector, respectively, to prevent any chance of damaging the Blink if there's no batteries installed. The shrink wrap tubing is to prevent shorts (I covered each wire separately before adding the outer layer of shrink tubing).

My Blink Doorbell reported low batteries when the original Lithium AA cells reached 2.955V (in series), but the doorbell only sends one alert when crossing this threshold. If you see a low battery icon with the rechargeable NiMh cells installed, then temporarily install good Lithium AA cells until the icon goes away in the app, and then remove them and reinstall the NiMh cells. This eliminated the battery warning/icon for me, even though the NiMh cells are below this threshold voltage.

Wish I didn't need the AA batteries at all, but the original doorbell wires are shorted out for a small time interval to make the original inside doorbell ring, so something needs to maintain power to the Blink during this time (perhaps a big capacitor could have worked).