For a fight, you want:
A place - where is Azzy most vulnerable, and how do you bait him to go there? He's smart, so he'll know it's a trap, but he's arrogant.
A time - Maybe some allies? A diversion somewhere that draws away some of his guards or allies?
A tool - I assume he's immune to most normal weapons. He's not going to go down to typical spells either...his reserve of power is just too great. It's like fighting the BBEG from Worm - a straight fight does nothing, he can last longer than anyone else. It could be a sword, or an epic level spell or ritual, or a poison made from the willingly given blood of the lords of the 8 other circles, or _________.
A weakness - Where are you putting that tool to use?
For not a fight, you want:
A philosophy - How is the party different? What are they using to attack him that nobody has ever used before? How can you target not just his body, but his belief and spirit? What idea could break him or cause him to give up? This may need to be something that completely subverts the expectations of a final boss battle. Maybe you can redeem him, or get him to quit in favor of someone more competent, or offer him "rest" via a ritual that kills him but seals off 1/3 of the Abyss, or something like that.
Example: In the Bible, Jesus doesn't show up with a sword and armor and duel Satan in an epic, mountain-rending swordfight. Instead, he humbly sacrifices himself so that all who believe in him are counted as clean and justified, despite their sin, and thus Satan has no power over them any more. Everyone, even his disciples, thought the kingdom of God would be a literal earthly kingdom, and expected him to kick out the Romans, because they were thinking on a purely material level. For something like this, killing one of the primary and most powerful figures, one who filled the Lucifer role of getting a bunch of angels(devas/whatever) to defect, a purely material solution should not work, and should indeed prove disappointing.
There was a story hour here a decade or so ago that dealt with philosophical/idea based approaches as well. I'm drawing a blank on the name, but it involved a Paladin trying to redeem a succubus and free her from Grazz't. I think he ended up partly succeeding and partly falling but also some of the gods quit or changed alignment, the rules for summoning in the material plane got changed, and they ended up in high epic levels.
They may need to make some Religion checks to start in the right direction. They may need to also take some sidequests to look up the History of Asmodeus (there are multiple stories, all partly true and partly lies, that he's given out) to try to figure out the real truth and what they may need to do. I'd include a trip to some Upper Planes to do research and maybe test out their arguments.
Of course, this all becomes less "D&D, roll d20 to hit" and more free-form....
Warning: This can get very complex. You don't have to figure out the solution; you just have to get your players thinking in the right general direction, and then if they come up with something that sounds really good, it Just Works. Let them unleash their creativity.