Assuming you have hooked it up correctly my next thought would be to make sure that you have match the default baud rate of the ESP8266. Also check the reset line I think it needs to be pulled up.
I would forget the reset pull-up and pull it to GND momentarily and see if a reset gets you back to where you were able to see the LED. If that works, I would get a 3.3v compatible FTDI board (if you don’t have one already) and keep your attempt to talk to the board via AT commands simple by using a serial console instead of an Arduino (for now). A good FTDI board is a good tool to have in any case. With it connected correctly you should be able to see if the ESP board is booting and/or putting anything out on the console.
Search around the web and you should be able to fine examples of how to connect the ESP to a FTDI board. Hopefully your board in not damaged, just in need of a reset.
I only have ESP-01s. If I had a 12E module I would try to do what you are attempting to help figure this out.
BTW, my ESP-01 provides Wifi for a Mega 2560 that talks to ThingSpeak. I know that when I originally set it up I was able to send AT commands to it and that I most likely initially used a FTDI board (3.3v) to talk to it.
I would get a good 3.3v capable FTDI board. You should be able to see if the ESP-12e is trying to boot with it. Also, if you need to re flash it the FTDI will come in handy. There are many versions FTDI available from very inexpensive to reasonably priced. You will not regret having this tool. I use the FTDI Friend from Adafruit. Sparkfun also offers FTDI boards and many others go for cheaper ones. It’s well worth whatever you spend to have one of these at your disposal. Just make sure it is a good one (read reviews).