That is a good question, but of course it only limits the issue before a Christian Fundamentalist's dilemma perhaps.
In Orthodoxy, Catholicism, and in mainstream Christianity, "praying for one's recovery" is not a request for miracles. In fact, I know from experience, that in Catholicism (at least) the administering of medicine or even the scientific discovery and development of new treatments can often be viewed as an "answer to prayer." Seeking miracles, though often declared as a requisite by some "true believers," is not elemental to all theologies or Christian traditions. Some even see such things as unnecessarily "testing" the God they worship.
A good friend of mine was a professional Catholic theologian. Quite aged now, he is mostly retired, but occasionally gets called on by both clergy and even the USCCB from time to time. I was quite surprised at the dichotomous approach he has to the work he does. I would describe it as "shocking to the conservative religious" because in academic work, even when the result is religious, there is great importance that he places on "never having the end result tainted by superstition," as he put it.
Of course, were it not for meeting people like him I would probably have a similar view to yours. My experience in the JW religion left me both very sure of my reasoning skills but naive to the reality of how little I really knew about things. One cannot jump to conclusions on how religious people incorporate their scientific side into the holistic whole that ends up being who they are without meeting each and every one of them. Some of them are actually quite divided and prefer it that way to keep them going too far one way or the other.
So their dichotomy is actually cherished and kept in place at all costs. Science and other critical analytical forms of reason are often employed purposefully by many theists because of the fact that they will not easily or ever reconcile with religious though. They want a fail-safe to ensure that their convictions are more than mere credulity.
How many of us have a similar fail-safe in place to ensure we are not catering to our emotional need to be right, have all the answers, and are now in a position to remind others with opposite views that they need to change their ways (again, for those of us who thought this way as JWs)? Just because we chose atheism and science doesn't mean we don't need to constantly ensure and validate our current stand with just as much discipline as these people do.