That's surprising to me! I thought that if Что is rude here, then что же must be more rude.
What if you began the sentence with почему?
Почему (же) вы вовремя не писали, на приёме?
Is there a difference in politeness here, if you use же, or don't use it?
P.S. Note that I edited my previous post a little bit, before I saw Doomer's response, but my edited version also used the construction "Why the heck?" to translate Что же, so Doomer's point is still the same.