Flight Instructor reported an engine failure with attempts to restart resulted in an off field landing.

Narrative

Approximately XA:30 picked up IFR clearance from ZZZ1 clearance delivery (cell phone); took off within 6 min clearance void time. Approximately XB:20 started VOR/DME XX into ZZZ2. Completed full approach; broke out a few seconds after holding at MDA (1520 ft.); touch and go [Runway] XX to missed (amended by ZZZ Center) (published but contact [frequency] XXX.XX; not previous controller at YYY.Y). Went missed; at 2;000 ft indicated; called Center; no answer...2;500 ft. no answer... just prior to 3;000 ft. we finally heard them...they asked if we could accept 5;000 ft; we said affirmative...they said pass 3;000 then direct to ZZZ; climb 5;000 ft. Passed 3;000 ft turned direct ZZZ; I felt/heard what I believed to be engine icing...checked rpm gauge (indicated normal operations) and asked student; he did not concur with my assessment...I pulled carb heat and monitored for a minute or two...no change. I put carb heat back in and thought I might just be over-reacting. We were approximately 3;400-3;500 ft indicated and started to lean mixture...engine died; thought nothing of it (mix was pulled too far out; so just put back in and try again)...but after going back to full rich; engine did not start back up. Cycled throttle; cycled mixture; nothing...carb heat out; mix full rich; throttle full in; went to right tank (waited a few seconds); nothing...left tank; nothing...mags were on both...turned left mag; nothing; right mag; nothing...even went to start just for the hell of it (prop never stopped windmilling)...nothing. Notified Center of engine failure; they did not respond; I asked again...told student to pitch for Vg. I ran engine out checklist line by line; still no start. I took control; told student to work COMMS (I don't recall much; he was talking to a relay aircraft and maybe Center); he squawked 7700 (I didn't notice; but he said so). Told student when we break out; I look left and he look right...first one to spot something suitable call out...we broke out at roughly 1;700 ft indicated (900-1;000 ft. AGL). Shortly after he called out suitable site...I swung around airplane; got in sight and started to dial in he continued on COMMS; I did one last ABCDE [checklist] seconds before contact I secured the engine (all fuel; mags; throttle; master off); prop ceased to windmill. After composing myself; tried to call Center on handheld...no luck. Called Clearance Delivery to alert them...they gave me Center's number...I gave them all the pertinent info and our GPS coordinates and awaited authorities.