It's a matter of satisfying the conditions. Dealing the damage is not any kind of action, it just happens. So any time the trigger condition is met, it happens. There is nothing that indicates if you need to rewind time and check to see if it already happened during this turn. It doesn't say once per turn, it doesn't say as an X type of action.
"Until the end of your next turn, any enemy that ends its turn adjacent to you or that hits or misses you takes 5 radiant damage."
Enemy A moves adjacent to you and readies an action to attack you when his buddy flanks. That ends his turn and he takes 5 damage. Enemy B flanks you, enemy A's readied attack goes off, he misses you and he takes 5 damage. Enemy B attacks, hits you, and takes 5 damage. Enemy B's turn ends, and he takes 5 damage. Enemy C walks up and attacks you, hits, takes 5 damage, pushes you 2 squares. Enemy C's turn ends, but he is not adjacent to you, so no damage.
That's about the fairest interpretation without injecting any assumed rules text.