So I tried to convert the alchemy lab in Vlindrel Hall to a child's bedroom. I went to the keep, paid the gold, but it never converted.

I went back, and I was able to pay the gold AGAIN but yet again it didn't convert.
I DO have Dawnguard, and I'm wondering if this is related to the same sort of bug in Breezehome.

Has anyone else encountered this problem, and have they been able to figure out a fix? Thanks.

It could be the same bug as in Breezehome (started a new character so got that one) but I haven't played Markarth in a while so can't confirm. You could try not loading Dawnguard to see if that will let you buy the upgrade then add it back in afterwards. That worked for me in Breezehome when I tried but I hadn't started any Dawnguard quests. Make it a new save in case it causes trouble.

I might give that a try. It will probably be a nightmare though, I have Serana as my follower and I'm all the way through the main Dawnguard quest (and I have a few sidequests up currently).

Hmmm, could be risky. There was a mod fix for the alchemy table out so I guess one for that issue is not far away. It looks like it's affecting all city houses in some way or another (judging by recent posts). Sorry I can't be more help.