How long does it take for a contrail to hit the ground?
A contrail is a cirrus cloud composed from water vapor nucleating on soot in the engine exhaust and the contrail itself doesn't fall to the ground.  You will occasionally see cirrus uncinus "mare's tails" form from contrails, which are heavier ice particles in the contrail precipitating from the contrail.  These particles evaporate long before reaching the ground.  

 
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The fate of the contrail cloud itself is death via entrainment.  Watch some over time as they distort from the winds aloft and then mix with the surrounding dry air and evaporate.

Once the contrail is gone and only vapor remains it moves with the air it resides in.  For synoptic scale flows, vertical motion of the atmosphere is on the scale of centimeters per second and as such until that vapor finds its way into subsiding air (e.g. a high pressure system, the dry conveyor belt of a midlatitude cyclone), it will tend to remain aloft.