@azygous is certainly much more knowledgeable than I am when it comes to issues chicks can have. She makes a good point about the eyes and possible internal issues.
They will sleep in a heap so the splash should be fine for heat.....do listen for loud peeping though.
I have had the occasional bird that stuffed themselves and had a huge crop. It looks to me like she just made a pig of herself. Being sleepy may be because she ate so much.
With solid sides on the brooder and where the heat lamp is it is possible she did indeed get to warm. Solid sides hold in...
Looks like she needs to rest and digest. I bet she will be feeling much better in the morning.
Leave the heat lamp off and see how they do. If they start peeping loudly check on them of course. I bet they just go to bed and sleep quietly.
Can you go in, turn the room light on and take a few pics.
The brooder itself so we can see where the heat lamp is currently and size of the brooder.
Yes pics of the chick will help us to help you. Just 2-3 pics are needed. The chick itself whether standing or laying and that crop.
As long as they are active when the room light (not heat lamp) is on they are ok.
Piling up to sleep in a heap is normal.
Peeping loudly and constantly is a sign of distress. If they aren't hollering all is ok.
Pics of the new poorly feeling bird? Try for a pic of the whole bird standing or...
With any animal that has been without food for an extended amount of time YOU have to slow them down by allowing small amounts frequently until they have recovered.
Take the dish out now. Let her rest (and hopefully digest a bunch of what she wolfed down).
Give three tablespoons once an hour...