Dukkha: Suffering
Dukkha is the ache built into conditioned life. "All our affliction" and "any affliction" cover the same wide ground, from the burden in Asia that left a man despairing of life down to whatever is pressing on the reader today. No kind is ranked above another here, which is why comfort can travel across kinds, the way water finds its level in whatever vessel it is poured into.
Assāsa: Consolation
Assāsa is relief, the easing that lets a person breathe again. Comfort arrives in the affliction rather than after it: the load stays on the back while the breath comes back. A fever breaking leaves the body still weak, and yet everything in the room changes.
Karuṇā: Compassion
Karuṇā is the heart's trembling at another's pain. Having been comforted first is what tunes it, so that the same darkness is recognizable on a stranger's face across the room and the feet move toward them. Whoever has been pulled out of a ditch knows where to put their hands.
Kalyāṇamittatā: Good friendship
Kalyāṇamittatā is companionship that carries a person along the road. "That we may be able to comfort those who are in any affliction" turns received consolation into equipment for that road, something meant to be spent on the next person who stumbles. One lamp lights another and loses nothing; whoever sat with you last winter is the reason you can sit with someone this one.
Kataññutā: Gratitude
Kataññutā is knowing what has been done for you. "The comfort with which we ourselves are comforted" keeps the source in plain view, so the comforter is handing on borrowed goods and stays aware of it, which is what keeps the hand open instead of closing around the gift. The Pali word has a twin, making the kindness known, and that is the whole motion of the sentence: comfort received, comfort passed along, nothing kept back.