It sounds like they might have struggled to get someone else in, and apparently Jardim was still bumping into his players in the bakery and within the locality so they probably felt the relationship wasn't too fractured for him to be asked back. Financially, he's got a payoff and another salary from them!
But for Henry, those Renault ads are going to be used endlessly in the banter when he inevitably returns to Sky Sports next season but quite early on it seemed he wasn't a head coach material. He was telling Falcao how to call for the ball (Moyes-esque with him showing Ferdinand a video of Jagielka) and he doesn't seem to have that warmth to relate to players. Maybe he'll come back into the game as an assistant coach, or a specialist striker coach but not as the main man.