Well you did just limit the movie to an absolutely ridiculous window of 36 hours, so that Jarvis who may have used heat scanners is the same Jarvis who by your time frame was just malfunctioning a couple of hours prior. Also, let's not forget the fact that Tony Stark now lives in a world where he know that aliens can open wormholes at any point in time allowing an entire legion of hostile soldiers to come pouring into our world. It was impetuous and stupid and unfortunately it kinda ruined my enjoyment of the movie as a whole. I would still rate it above Iron Man 2 but as far as the current batch of Marvel Movies go, that's it. Cap, Thor, and Incredible Hulk were all better overall in my opinion.
Jarvis was rebooted when the suit regained power
Just because his speech program was malfunctioning doesn't mean that his other operators weren't working properly, especially since we just witnessed a huge battle in which the AI was functioning properly