For what it’s worth, all of the example sentences in my dictionary that relate to a person currently being sick use ~ている. For example:
彼女は心臓を病んでいる。She is troubled with heart disease.
Sometimes in Japanese you’ll see a present condition being described as the past tense of a verb that’s about becoming that condition, e.g., 疲れた (I became tired) for “I am tired”. But in this case it looks like the present condition is formed as continuous, i.e., ~ている. I’d guess that the present tense (病みます、病む) is the “becoming” sense, as in 病気になる.
All guesses though, hopefully someone will have a more authoritative response.