、。
Perhaps sensing he couldn't be persuaded, Kurosaki-sensei sighed and took his seat.
Literal
Persuasion-impossibility [object-を] sense-[and-て][question-か], Kurosaki-sensei [topic-は] sigh [object-を] let-out-[and-て] seat [to-に] attached.
The ~してか construction ('-ing or something, perhaps because -ing') is a compact way of speculating about someone's motivation — literally ending the subordinate clause with a tentative か. 説得不能 ('impossibility of persuasion') is a crisp kango compound. Two idiomatic verb+object pairs: ため息をつく ('let out a sigh,' literally 'put out a sigh') and 席に着く ('take one's seat,' literally 'attach oneself to the seat'). Together they paint the scene of resignation.