You may not realize it, but this is actually two different questions that have two different answers.
It stopped being about his job as soon as he learned about Ashley's (and his) plaga.
It became about Ashley after the Mendez fight, when she helps him to his feet and he smiles at her for the first time.
When Luis told Leon and Ashley about las plagas, Leon had a very important decision to make: he could accept Luis's help OR he could turn him down, stand by and wait for extraction, and take his chances with US government and military doctors in hopes that they'll figure out what Luis claims to already know.
By all rights, Leon should've told Luis to piss off and put his faith in the military. Any other agent would have. It only makes sense to trust American military intelligence to know what they're doing -- especially when the alternative is a shady-ass total stranger in a foreign country who was probably involved in some pretty sinister shit in the past and has already left Leon to die once.
In fact, Leon in particular should have made the decision to turn down Luis's offer and just bring Ashley home, considering his history with Umbrella. And, after all, his mission objective was to find Ashley and return her to the US. Nothing more than that.
But that's not what he does. Instead, Leon makes the conscious decision to ignore his mission objective and stay in Valdelobos longer than he rightfully should have in order to place his trust in a man that he has no reason to trust -- except for the fact that Leon knows, from personal experience, that the only people capable of cleaning up Umbrella's messes are Umbrella's own scientists. He saw how hard Annette fought to not let the G-virus spread past Raccoon City, and I'm sure that Claire told him about what happened with Sherry.
So, it stops being about his mission in that moment that Leon decides to disregard his orders and trust Luis. It stops being "just a job" right there, and it starts being about Leon's baggage about Raccoon City. Because not only does he deviate from his mission objective, he also does not report Ashley's infected status to Hunnigan at any point. Because he knows that if he does, he'll be ordered to stand down and await extraction -- to do nothing and wait for US doctors to look at Ashley -- and his hangups about RC can't handle the thought of doing that.
His desire to rid Ashley of las plagas becomes about Ashley herself as a person after the Mendez fight -- after the second time that she, instead of hiding and waiting for him, came back for him and saved his ass (the first time being the cabin). And not only does she come back for him, but she smiles at him and offers him her hand.
The sad reality is that this is probably the nicest that anyone has been to Leon in six years. Ashley shows, through her actions, that her concern for him isn't just born from the fact that he's her rescuer; she genuinely seems to care about him as a person, during a time in Leon's life when no one else does.
And after she extends her hand to him, he smiles at her for the first time. It's actually the first time in the whole game period that he smiles at all. And this is the point in the story where Leon's dialogue to her changes dramatically in tone.
Prior to the Mendez fight:
Ashley: What's... going to happen to me?
Leon: Right now, let's just focus on getting out of here.
Ashley: I'm not going to turn into one of them, right?
Leon: I won't let that happen. I promise.
Almost immediately after the Mendez fight, Leon starts playfully teasing Ashley. "Well, I wanted to go home, but Ashley just had to see this castle first." and "No, you can't keep him" re: the garrador and "What are you, my mother?"
And also in the immediate aftermath of the Mendez fight is the "The girl's just fine. With me." moment.
Like, this is really where Leon's focus changes from general anxiety about Umbrella and las plagas to Ashley's personal well-being. Ashley extending her hand to him isn't just an offer to help him up; it's symbolic of her reaching out to him as a person. And when he commits to taking her hand, it's symbolic of him opening his heart to her.