How to recover files from a phone with a broken/unresponsive screen running Android Marshmallow? I recently dropped my Nexus 5 and the touchscreen stopped working, no matter what I do it won't function. I have files on this phone that I didn't backup (I know, real smart of me), and I'm wondering how to get them back using my computer and a USB cable, considering it's the only way possible since the screen isn't working.

The catch is that with Android Marshmallow, came the feature in which when you connect your phone to a PC, it won't get recognized right away. You have to unlock your phone and swipe down the notification tray and tap on a "USB Enabled for Charging" (Or something similar) and select "USB for File Transfer" (Or something like that...). Since my screen is broken, I can't complete the aforementioned process.

What can I do to recover my files? The screen is completely broken, it registers some touches when I put pressure on it but it's just random taps all around the screen that make it go crazy. I have no pin or any form of security enabled, so I can just press the screen a bit and it'll unlock (with some luck and several tries), but beyond that any action is impossible to realize.

Thanks in advance!
Turn off phone using power button.
Boot into recovery mode. 
Connect it to the PC and most probably it would be connected as MTP. If not, then recover your files via adb. 

Edit: As pointed by @Firelord this will work on custom recoveries only. 
The other method is as follows. 
If you're connected to a Wi-Fi network, then find the IP address of your phone, most likely from the router setup page. You can then use ssh to copy files from the device. If ssh is not an option, then boot into fastboot mode. Flash a custom recovery for your phone and then follow the original answer.