last-minute speaking engagement; sweeping staircase

A sweeping staircase is a grand set of wide and curving staircase usually in lobby of a large hotel or a grand foyer of a mansion.

A speaking engagement is when someone gives a lecture or academic level talk to a group or association.

"who is this engagement for, me or her?" There isn't enough information in your post to answer this question. I need more context to figure that out.

Thank u! Actually the speaking was talking to her pet about her arrangement and "speaking engagement" just appeares out of the blue. The part that follows this sentence recounts her travel to meet her friend. So I guess even if i enter a whole page of information here, it won't help us to find the answer.
Many thanks, anyway!