I can see some drone brood n the "queenless" colony which suggests she is either failing and unable to lay fertile eggs or you have a laying worker. This happens when there is no brood or queen pheromone in the hive to supress he workers from developing ovaries. It is very difficult to requeen a laying worker colony as they often kill the new queen, but they are doomed without one as the workers can only produce drones ad they are small ones anyway, because they are developed in worker cells. The colony needs workers to exist, so they eventually die out. Yes they will be producing honey because they have very little brood to rear... which eat up the nectar and pollen they forage.
The best thing to do is to tip/hake all the bees out of the hive and off the comb 40- 50 feet away and put a new hive box on the old site to catch the returning bees. At dusk, tip the bees from that box onto a ramp leading up to the other weak but queen right hive and hopefully they will integrate and make that one stronger.
I see 2 eggs in some of the cells = laying worker.