Just to follow up.
I was suggested to try out this implementation for MPU6050 (instead of reading and combining data using Kalman filter): http://goo.gl/A1W2uP
After I implemented that I got far better results than before, still not balancing but I haven't had the time to play with PID tuning yet. I plan to add a running average filter and place the MPU a little closer to the base of the robot. I hope that it will improve stability and the robot would finally balance. Here is a clip of current progress, even though the robot is still not balancing: Dropbox - Error - Simplify your life