Why Air France 447 could not reduce angle of attack?
I just finished reading this article and couldn't understand why in 3
  minutes and 30 seconds even though there was no reliable speed data,
  pilot could not level the plane?


If you look at this video by the French accident investigation bureau (BEA) you can see what happened.

The problem started when the speed indicators failed temporarily because of icing of the pitot tubes. The aircraft was flying through the intertropical convergence zone (ITCS) where severe thunderstorm activity can build up to high altitudes. 

Instead of maintaining attitude and thrust setting, which is a normal procedure when speed indication is lost, the pilot flying (PF) brought the nose up. The aircraft started to climb but at the same time lost a lot of speed (trading kinetic energy for potential energy). Eventually the airspeed dropped so far that the aircraft stalled. By the time the aircraft entered the stall, the pitot tubes where de-iced and all speed indicators were working again. From the moment the aircraft stalled it started to sink. The PF tried to maintain a nose-up attitude, eventually using full aft stick. We can only guess to the reason why; it is likely that he was disoriented and had no confidence in the instruments after the failure. Yet it goes against basic flying instincts to pull the nose up in a stall. 

The aircraft is naturally stable and will automatically drop its nose in case of a stall so it took some effort by the PF to keep the nose up. The horizontal tailplane then stalled as well which removed a lot of the nose-down tendency. The result was that the aircraft was in a nose-up attitude for the most part of the descent.  

The angle of attack remained therefore too high and the aerodynamic stall persisted all the way down. If the crew would have brought the nose down they would have increased the speed, lowered the angle of attack and recovered from the stall. Once recovered from stall they could then level off the plane.

It may have contributed to the confusion of the crew that the airspeed was below 60 knots during portions of the descent. As they accelerated through 60 knots, the stall warning re-appeared. 

The stall indicator systems makes use of alpha vanes which need some airflow over them to work. By inhibiting the stall warning below 60 knots the system is prevent from giving nuisance warnings on the ground. Such a speed is not normally encountered during flight, even not during a stall.  

Another contribution to the accident may be that the pilot-not-flying (PNF) cannot see what stick inputs are given by the pilot flying. If it would have been in a different aircraft with coupled yokes, the PNF may have realized what was going on. 


  Isn't the angle of attack reported by a gravity based tool like in
  off-road vehicles?


The angle of attack is the angle between the chord of the wing and the incoming airflow. It is not related to gravity. 


  Summed up Question: When it's unclear what is happening with the
  aircraft, isn't it safer to put it in level position and medium speed?
  (Sorry for rudimentry language here)


Initially only the speed indicator failed. If the crew would have remained at the same attitude and power setting the aircraft would not have stalled. That is the standard procedure in such a case ("thrust and attitude")  That may not have been the proper reaction in this case. The aircraft was encountering severe turbulence and the fastest way out may have been to climb. But after the stall warning sounded all speed indicators were back to normal and the aircraft was flyable.