Based on the context of the sentence, the most appropriate word to fill in the blank is "amusing".
The sentence suggests that the speaker is apologizing for being late and hopes that their sister has been keeping the person who was waiting entertained or amused in their absence. Therefore, "amusing" is the most appropriate word to convey this meaning.
Neither "tickling" nor "playing" would be appropriate in this context, as they do not accurately convey the idea of keeping someone entertained or amused. "Tickling" suggests physical contact, which is not implied in the sentence, and "playing" is too vague and does not provide enough context for what the sister might have been doing.