Some of you guys are trying to find a technological answer to an emotional problem. The scene isn't specifically about the cure they found later ('weeks' is never mentioned). That they found the cure not too much later (again, 'weeks' is never mentioned) is only meant to be the final nail in the coffin to McCoy's burning guilt about his father.
He could have said months, he could have said years, but he didn't, because the length of time is only secondary to the actual emotional cost that they did find a cure, and McCoy's survivor's guilt, doubly compounded by the fact that he was not only a doctor, but his son.
So you have a bright, young doctor, and his ailing parent. Doing everything he can to keep his father alive, desperately, and his father is so tired, so exhausted from all of the pain, from all of the life extending treatments that aren't working, he reaches out to his son, who is his doctor, the only one who can release him from the pain, the one he trusts to do this, and his son is conflicted, yet because he loves his father, because he doesn't want to see him in pain, and because there has been no word of a cure, he agrees.
Then, some time later, a cure is found.
That is the crux of the whole scene. How long it takes doesn't matter so much beyond the fact that it wasn't "too long after", which is nicely vague enough to make the impact of what happened, while not tying down too many details.
In short, you're asking the wrong question.
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