iPhone won't turn on, but will connect to iTunes

So my wife's iPhone recently "died". An alarm started going off and she went to turn it off, but when she did the screen just stayed black.

I followed some of the guides and other answers here and tried a hard reset, then tried connecting it to the computer. This is where it gets weird.

The phone connects to the computer with no issues whatsoever, the computer can pull data off the phone including serial number and all that. It was able to update the phone even (I attempted a recovery once it seemed like there were few other options).

However, when I try to interact with the phone it just stays on the black screen. I tried getting Siri to activate with no luck, and when I move the silent switch it doesn't vibrate like it normally would. This makes me think it isn't just the screen needing to be replaced.

My initial thought is probably the logic board, but I'm not sure the logic board could be the issue if it is able to connect to the computer with no problems.

Thanks.