I have accepted Lord Ironfist's quest to find Lord Kilburn, and I find him. Lord Kilburn, in turn, then wants to send me on a quest of his own, it seems. Whether or not I accept Lord Kilburn's quest, when I return to Ironfist, I can't ever seem to get credit for finding Kilburn. I feel like the Ironfist quest chain is broken now somehow, or am I missing something?

IIRC it's not a real quest, but more of a way of steering you along on the main quest.
Did Ironfirst give you another quest, like finding an item or killing a monster? If so that is the "real" quest.

No, it is a real quest and Lord Ironfist should reward you for finding Kilburn.

However Lord Kilburn doesn't give you any quest (he just gives you a helpful item).

So the question is: Are you sure it is Lord Kilburn you found? Please note that Kilburn is not a lord in a castle but a poor guy somewhere outdoors.

Thank you for your responses.

Yes - I have indeed found Lord Kilburn and he gave me his special item. Upon finding Kilburn, I have the opportunity to say Yes or No to his quest. I initially said Yes, which I fear may have broken or overrode the Ironfist quest. I then had my Cleric cast Remove Quest and tried with Ironfist again. Accepted the "Find Lord Kilburn" quest again, but still no dice. Very odd.

In the meantime I have been assisting Inspectron so there is no lack of things to do. Just kind of annoying since I am a bit of a completionist!

Thanks again!

That's really weird. I don't remember having trouble completing this quest.
By the way, accepting his map doesn't interfere with Lords' quests. I always did. And the quest he gives you isn't a real one.

I searched for walktrhoughs, and one of them says "For finding Lord Kilburn, your leader needs to have the Map of the Desert readied". Maybe you should have the map equipped or something when you talk to the lords.

This might be it... I would at least make sure the party leader is the one carrying the map. But, like ZFR, I don't recall having any problems completing this quest, so it's a bit odd.

I'm glad that worked! The game is so old that there's not much data that actually gets saved, so it uses items to track quest progress. Which can unfortunately lead to weird situations like this one.

My goodness this was a frustrating experience.

I tried cancelling and retrying the quest in various ways around 6 times over before I found this post. Traded the map to my party leader and finished. Thanks you guys across the years.

No problem. It's generally a good idea to have the leader carry quest items, there may be other situations where this happens also.