I just tried to set up a brooder...My bird would not stop yelling even though I checked the temperature and it was around 100 under the lamp and about 85 on the other side.
I ended up feeling really bad and took her out and set her back up with a blanket cave and heated rice sock and room for her to come out on the other end of a shallow cardboard box where I have a tiny cap of water and a small amt of food on each corner. The problem is, I have to wake up and re-hear the sock about every 3 hours. The chic seems very content and. Ever cries with this method!
The blanket and the rice sock are something cozy to snuggle against.
Maybe you can put them IN the brooder, about halfway between the warm and cool areas, and tuck the chick in there. Then don't bother heating the rice sock anymore, but leave it as a comforting "mother" for the chick.
Considering the temperatures you measured, I would probably try a smaller light bulb for heat, because the chick probably does not need the warm side quite that warm, and it would be better to have the cool side even cooler.