Most of it has already been said, but a few things in your posting still can be emphasized:
swordsmen and horse archers are completely irrelevant against longbows. Note that they have the same base strength as longbows... and almost always the longbows will have a LOT better bonuses.
against longbows 2 catapults is not enough. bringing down the walls should be followed by "suicide siege", siege weapons that bring collateral damage and put a dent in the top defender.
trebuchets make it easier. But that doesn't mean you need to wait for trebs. It's perfectly possible to defeat longbows with macemen, catapults, possibly knights, and more catapults.
lastly: this is all possible, but still tough. So anything that makes it even tougher means you need a seriously large army for it. If the city is on a hill or the leader is protective, expect to need multiple suicide catapults.