Sparks and no power after accidental short
You smoked your neutral

Now with your pictures, I see this is not a multi-wire branch circuit, just a simple circuit.  

The white needs to go to neutral, and from your pictures and comments, we can see it does not.   I would bring it to a different screw if it will reach; this one may be charred.  Still, loosen up that screw and get the bit of wire out of it. 

I suspect it "burned out" at this location, right where the wire meets the screw.  Perhaps the screw was loose, and that caused arcing which damaged the wire over time; or your short was enough to cause the burnout. I suspect a little of both. 

The black needs to go to 1 circuit breaker, and the breaker needs to be an appropriate size for the cable.  The cable will be marked either "14 AWG" which needs a 15A breaker, or "12 AWG" which needs a 20A.   If the breaker size was wrong, that would contribute a lot to the wire burnout.  

Neutral being dead explains why the non-contact tester senses voltage on the hot ... but the conductive tester doesn't sense any voltage.  As you know, a voltmeter requires 2 connections and the neutral is dead.