I'll have to _________ because I don't know how Sheila's parents are going to react to this offer.

The speaker of this sentence wants to be adaptable and/or spontaneous because they are unsure of how someone will react. Therefore, the blank should be filled with a phrase that expresses the idea of responding to a situation spontaneously, without a set plan, based on how things turn out.

The idea of reacting to an unpredictable situation is best suited to "play it by ear" (Option B), which basically means to improvise or act without a plan, depending on how the situation unfolds.

The other choices don't fit as well. "Count my fingers"(Option A) is quite literal and refers to counting with the fingers, which is inappropriate for the situation. Accepting the results of one's actions is what it means to "face the music" (Option B).However, the sentence is about anticipating an uncertain response and ignoring the repercussions. The phrase "break the ice" (Option C) refers to striking up a conversation in a socially awkward circumstance. This also doesn't align with the idea of waiting to see how other people respond.

Thus, option D is the correct choice.