"During" indicates that something is happening at the same time as some other ongoing action. Notice that this includes the implication that something is actively happening. So you might say you were doing something during the baseball game. For example, "I was knitting a sweater during the baseball game." But you would not say that something DIDN'T happen during the baseball game, because something not happening is not an ongoing process, and does not require any time at all.
"I couldn't do anything" is a simple statement of non-action; it requires no duration, and so it does not require the use of the word "during." "For" is the way we summarize the entire period of time (the rest of the day) in a single lump, without emphasizing the ongoing duration of it.
To reiterate, the two words indicate a difference between action and non-action, but even more importantly, they also refer to different ways of looking at the period of time involved. During, which really means "within the duration of," typically refers to an event or an action (such as "the baseball game"), which occurs within or delineates a period of time, rather than any period of time itself. "For" refers to some period of time itself ("for the entire year," "for a whole week"), rather than events or actions.