Phone dropped, not responding, but update goes through. Can't backup

Hi,

Our baby dropped my wife’s phone quite strongly. We have no backup! The screen went blank with backlight still on. I tried to reboot it but couldn’t do soft reboot as nothing appeared on the screen (couldn’t slide I guess). So I did hard reset with volume down+power button. The display turned off. But now the phone wouldn’t turn on (I think).

If I press the power button nothing happens. If plugged into the computer with iTunes on, I am successful getting it into recovery mode. Nothing on the screen, but iTunes says that there is a problem with the phone and I should either update it or restore. I chose update and it went through (according to iTunes). But the phone still wouldn’t turn on. I can’t see it on iTunes unless I start it in recovery mode.

All I want for now is to be able to take a backup. But this is impossible in recovery mode…

Any tips?

Thanks!