there came = was brought. Why assume that Elijah then sent it? It might have "come" as Holy Scripture comes to us to-day, though written in the past. It does not say a "letter" (which would be 'iggereth, or sepher, a book), but michetab, any writing, written at any time; probably a prophetic writing to be delivered at this particular time.
Elijah. Long since raptured (2Ki 2; Compare 2Ch_3:11). This is the only mention of Elijah in Chronicles.