This is pure speculation and just came to me:

Mr. Woodhouse has been a widower for some 16 years. I don't recall that we find out exactly how Mrs. Woodhouse died, but it's fair to say that she passed away due probably to an illness. It must have been a very untimely death too, as she had to still be quite young. I estimate Mr. Woodhouse to be in his 60s (I don't recall his actual age being spelled out. Am I incorrect?) He would have been in his mid-forties when his wife died. Mrs. W, having two young daughters (Emma was 5, Isabella could not have been more than 10, since she is married to Mr. Knightley's younger brother.) So I would estimate that Mrs. W was about 30-35 when she died.

All this to say that Mr. W's obsession with health could very well have to do with the trauma of losing his companion so shockingly early in their lives together. One could speculate also that, as someone who seems to have a very good and tender disposition, he might have become rather depressed (in a clinical way) over the event. This could account for his slipping into a form of mild senility and his constant worry that others dear to him might, too, meet an untimely death due to illness.

On a subconscious level, I think Emma might feel all this, and because she is a very good person at heart (very much like her father in this respect), she tries to fill this void in her father's life. She is a model of patience with him, if you examine it. Anyone who has had (in RL) to take care of an elderly, fanciful relative can relate to her experience, I think.