Based on our conversation at the meeting/potluck, it is much more likely that there are laying workers than a "recent queen", either old or new.
The consideration to try to requeen the hive was based on the described strength of the hive in numbers (worker bees covering combs) and that the colony will need lots of workers now to support brood production for the next 6 weeks, after which egg laying typically declines substantially or shuts down completely. Indeed, a younger queen might be better as it would be likely to continue to lay longer into fall, but there aren't a lot of recently mated queens to be had right now, so any mated queen that can be obtained may have to do.
If my interpretation of what was described is correct, laying workers are likely a major consideration right now in conjunction with an attempt to re-queen. For lack of time, we didn't get into the laying worker issue during the meeting.
I generally default to one method for dealing with laying workers: 1. Move the hive a small distance (it can be 6 feet away). 2. Shake the bees out onto the ground at the old hive location frame by frame, reassembling the frames in an empty box. 3. Re-distribute the frames and boxes to my other hives. 4. All the bees from the laying worker hive are now homeless and eventually beg entry to a queen-right hive in the yard, and the queen-right colonies reform them.
I believe that Mila's situation is complicated by the fact that she does not have a queen-right hive at the moment. Reportedly, it is also difficult to get a laying worker hive to accept a mated queen, which is why I generally don't try.
If I were Mila, and I found a mated queen, I think I would probably follow steps 1-4 above just as I described doing for a laying worker hive. Once most of the bees are shaken out, I would prepare a hive with one deep or two medium boxes of frames with the caged queen inserted in the top box where I can easily access her, excluding any frames that have laying worker eggs and brood. Then I would place this hive back at the original hive location for the shaken bees to return to. I think it should improve the chance of the mated queen being accepted. I would leave that queen caged for 4-5 days and then check back on it, looking for signs that the bees are accepting her. Feeding the colony may aid in queen acceptance. I'd leave her a bit longer if needed, and I'd replace the cork with a marshmallow for them to chew through when I think it's time for her to be released.
I think a nucleus colony with a laying queen and some frames of brood that the laying worker colony could be assimilated into would be a better approach, but that may be harder to come by than a mated queen.
Joe