OMG, I read What Katy Did as a child too! (Somehow, I got the abridged version…I don’t know what the unabridged has in it.)
Now that you mention it, her fall from a swing and subsequent paralysis and sudden recovery after several years does seem a little deus ex machina.
Her father was a doctor, and I don’t remember there being a lot of medical care, other than she was in a chair, the heat from summer affected her a lot, and right after the accident she was in a lot of pain. Her cousin Helen was also paralyzed, but I think hers was a degenerative disease.
Is there a paralysis condition that can reverse itself that suddenly?