Sorry to hear about your WiFi and battery issues! I know what it's like to have a disconnected, drained device and deal with its inconveniences. Let's get your thermostat back workin'.
If the Nest's battery level gets below 3.6V, it will stop connecting to your WiFi network. Can you check your voltage by going to Settings > Technical Info > Power and let me know what number it reads? If the voltage is above 3.6, you're good.
If the voltage is low, there may be a wiring problem and your Nest may not be charging properly. Unplug it from the wall, then plug it back in to see if that helps. If not, you may need to call in a professional
to check your wiring.
I look forward to your reply and am hopeful we can resolve this issue together.
-From Ai, Community Specialist.