da Vinci Jr. nozzle diagram
After watching the video, it looks this is a Bowden style extruder.  The extruder assembly is encased in the black box the OP mentions.  The assembly is user removable.  It appears they also include two cleaning tools


Small (0.4mm?) wire to clean the nozzle tip
Large (1.7mm?) rod for pushing filament down that is stuck in the cold end


It appears the OP tried accessing the filament in the cold end by removing the fitting at the top of the assembly without opening the assembly.  Unfortunately that caused the nut holding the fitting on from the inside fell off.  At that point he was in a bit of a mess.

I believe the manufacture's intent is to "ram" any broken pieces in the cold end through the hot extruder by using the provided rod and pushing while the extruder is on.  It seems that it should work but would not be easy to do if the filament broke off outside the extruder (which seems to be the case).

At the point he left it, it would seem that he would either have to do surgery or replace the extruder assembly.