I’d look at something like this to handle the connection to the battery and both 5v and 12v output: https://www.amazon.com/Dewalt-Adapter-Converter-Output-Battery/dp/B08GZ5M3YW

The 12v connection can power your LED strip and the 5v USB will run your ESP8266.

Also, rather than a reed switch you can connect a cheap motion sensor to your ESP8266. Opening the closet door will trigger the motion sensor and you can use that to programmatically turn on (and keep on) the closet light.