First you’re very sad! You are allowed to say rude words then and blame yourself.
As for the workers, they‘ll notice the disappearance of their queen within a few hours. If you listen very carefully the noise made by a colony, the humming or buzzing is different between a colony with a queen and a queen-less colony. They ‘ll know she’s gone because when the queen moves inside the beehive she deposits pheromones from under her feet. For the workers it’s some kind of “well being perfume”. The pheromonal bouquet of a hive is its signature regarding other honeybees, siblings not belonging to this colony.
Workers are not comfortable without a queen, they are stressed, so they‘ll select a few eggs or young larvae (less than three days old) to raise them as virgin queens and not as sterile workers. We don’t know why they choose this particular larva and not that one. One day maybe researchers will discover why.
The development of the virgin queen is quick compared to the development of workers or drones, but the egg needs three days to hatch, then the little larva pupates and the adult virgin queen emerges in 15 days.
Virgin queens will fight each other. A week later, the winner goes to a drone congregation area, mates with several drones and comes back to the hive.
One more week is needed for the queen to lay fecundated eggs. (Queens need the right temperature inside the group of workers for the spermatozoids to migrate to the spermatheca.)
It will work smoothly for the colony if there are still drones around… When the mating season is over, workers tend to get rid of all the drones, they ‘ll die of starvation and cold outside the beehives.
So, please, if you really have to kill your queen-bee, do it only during the mating season or the colony will raise a virgin queen but that will get you nowhere. The eggs that a virgin queen can lay are not fertilised, so they will evolve into drones only, not workers. No workers, no work: no colony…
While all this queen development cycle goes on, the last of the brood laid by the dead queen has emerged, so when you look at the frames of the hive, you see no brood (24 days from egg to emerging drones), and the next generation of foraging workers will start 40 days after the new egg laying.
Watch out not to miss honey/pollen harvesting in the meantime before killing a queen. If you want to bypass all that, you can buy a queen quickly and introduce it into the queen-less colony before the workers construct royal cells.