What is the best way to cool childs bedroom down, we don't have air conditioning?

A good fan

Don't use air con it's bad for respiratory system. Just get her a fan and she's good

Fan

Maybe set up a temporary bed in one of your cooler rooms downstairs

Fan or buy window unit

A fan with a strips of wet cloth hanging in front of it..

have a fan blowing on her, it won't cool the room but it will cool her.