I dont know how many days it would take, but i suspect over 3. There have been more than a few trout that i have pulled in to shore that have a hook or two still embedded in their jaw or throat.
Hooks that would be better to be 'left behind' would be steel, as it would rust.
Some of the newer alloys out there, including titanium hooks, would last a LOT longer.
If its a treble hook, then it would probably do more damage and kill the fish before it came out.
Other thought on this, how much line gets left behind? Cut the line near the hook, how many times will this tag of line get into the gills and kill the fish?