、、。
I'm having dinner with him tomorrow evening, so I'll sound him out then.
Literal
Tomorrow [possessive-の] evening, him [with-と] dinner [object-を] together do [nominalizer-こと] [dative-に] become-is [because-んで], that-time [that] indirectly sound-out try.
Packs several mid-level patterns into one flowing sentence: ~ことになっている for the established arrangement, the reason-marker んで (colloquial contraction of ので), the adverb それとなく ('indirectly, casually'), and the 'try doing' auxiliary ~てみる. 打診する literally means 'to tap/percuss' (as a doctor does a chest exam) but metaphorically means 'to probe someone's intentions' — a vivid piece of figurative vocabulary.