In the future timeline where Kes died of old age, she was in Medical permanently beforehand.

In general, the answer is probably exactly what you're thinking: if they need intensive care, they stay in Medical (probably in the swooshy stasis arch thing). If they're able to be released to their quarters, they do that - possibly with specialist equipment manufactured, and then daily house visits.

Though there's not many conditions that would cause long-term casualties. The Doctor would simply spend the episode researching it, and then fix it - or else they'd die. Trek-tech is very good at fixing the non-lethal.

(My phone does the same thing; you might try turning screen-rotation on and flipping it sideways, that often restores it.)

hS