The screen of the phone broke and you want to retrieve your contacts / files… Quite typical story. While getting your photos / files is quite trivial, plugging your phone to computer and copying necessary files would be enough.
The situation with getting out your contacts (if you happened not to sync them with Google) is slightly more complicated. Here is what I did in the case of Samsung S4 mini with broken screen. Note, the digitizer (touch screen) worked, but the USB debugging was OFF. Also, S4 mini has no video output. If your phone happens to have HDMI or MDL, just get the cable and plug it to your TV / monitor 😉

1. Enable USB debugging

This is hard part and can be done only manually! But it’s quite complicated with broken screen. You need to repeat 3 steps until you reach what you want: make a screenshot (HOME + POWER button in S4 mini), see what’s on the screen (navigate to you Phone storage > Pictures > Screenshots), do some action and repeat… This is extremely tedious, but proved to work with me.
In Android 4.4 which my phone had, you need to enter Settings > About, scroll down and press many times (~7 should work) Build Number. This will enable `Developer options` in Settings. You need to enter it and tick `USB Debugging` and press OK (here I needed to rotate the screen, as the right side of my digitizer didn’t work…).
I recommend clicking `Revoke access` and OK, as my computer couldn’t connect till I pressed it.
Then unplug the mobile phone and plug it again. New dialog asking for permission to access for your computer will appear on the screen. You should tick `Always allow access` and OK. From now on, the access through ADB is possible. You can check it with:

# install ADB
sudo apt-get install android-tools-adb

# connect
adb devices
adb shell

Note, if your digitizer is working only partially (my case), it’s usefull to enable autorotation first.

2. Screencast Android to computer monitor

# install seversquare
sudo apt-get install qt4-qmake libqt4-dev libqtcore4 libqtgui4
git clone https://github.com/yangh/sevensquare
cd sevensquare

# for Ubuntu 16.04 replace 5th line of Makefile with 
	(cd build && qmake-qt4 -o Makefile ../seven-square.pro)

# compile
make

# and run
build/seven-square &

Now, you should see the mobile screen and be able to interact with it your mouse & keyboard. Now exporting contacts should be trivial, right?

I have tried to dump userdata partition, but on original system version there is no root access and getting one would erase the data…

Let me know if there is any simpler solution!