I spent a year in the UK more than 4 decades ago. Those temperatures would seem hot to you but should not be hot enough to really bother the chickens. I don't think those temperatures are what is causing your egg production to drop.
The most common cause for production to drop is the molt...
It depends on what you use for a broody buster. I have a brooder built into my coop that is 3' x 6' (1 meter x 2 meters) with a 1/2" hardware cloth floor. If I'm not brooding chicks that is my broody buster. I've had as many as 3 broodies in there at one time. No problems.
I agree with the...