When to ask somebody else

Some answers are a no wearing politeness: a long pause, "I could write something brief", "you might get a stronger one from someone else". Take these at face value and thank them. A lukewarm letter does more damage than a missing one, because it is read as the best you could find.

Ask one more person than you need. Keep a list of who has agreed and what has actually been submitted, and check in once about a week before the deadline — short, no apology. Then tell them the outcome either way. People write these for nothing, and they hear how it went far less often than you would expect.