It is extremely unlikely that a person could experience major trauma to the heart, regain consciousness for a moment and utter something coherent, and then immediately die. As a person loses blood, or as their heart begins to fail, their mental state deteriorates due to lack of adequate perfusion to the brain. It would work much better for your story if the person were not aware and oriented, but instead just muttering the evil organization’s name as their awareness deteriorated.
In the field of emergency medicine, there are a few conclusive signs of death. The only sign that takes little or no time to set in is extreme and obviously fatal trauma. A stab wound to the chest does not fit that criteria, but a decapitation (for example) would. The other conclusive signs are stuff that takes too long to fit in your story, like rigor mortis, blood pooling, and putrefication.
On the other hand, there are lots of presumptive signs of death. If a person has no pulse, that is a presumptive sign of death, but not definitive. If a person has profound cyanosis, or is not breathing, that is another presumptive sign of death. If a person is unresponsive to any type of stimulus, including pain, that’s a presumptive sign of death. There are lots of other presumptive signs, like extreme blood loss, extremely low body temperature, stuff like that. None of these are enough for an emergency medical technician to decide that a person is absolutely dead, which is why they’re “presumptive” signs.
A person won’t stiffen up right away as they die, because it takes awhile for rigor mortis to set in. They might go slack as their central nervous system begins to fail, the heart might stop beating (causing blood from the heart wound to just ooze or pour out of an artery wound instead of squirting), they would certainly stop breathing, but even people who have absolutely no chance of ever coming back will do something called “agonal respirations” in which the brain stem sends a signal to breathe, causing an otherwise completely dead person to suddenly “gasp” once, in a way that I’m told is startling as hell to witness. If a person has agonal respirations, they’re dead (not officially, of course, but anyone who saw it would assume it).
I hope some of that info can help you write your scene. It sounds exciting!