Why is the copula היא obligatory here? The only notes section I found that talked about it was Adjectives 1 and it didn't really explain when it was or wasn't necessary. Would העוגה בשביל הילדים translate to "the cake for the children" or would that be העוגה הבשביל הילדים?
The translation "העוגה בשביל הילדים" is correct (you do not add definiteness prefix to conjunctions in Hebrew). "היא" is used in this context to indicate that this is a nominal sentence, but is completely optional.