I just reread your post and saw she died very soon after being moved to the brooder. Was she able to walk well enough to move herself to warmer or cooler spots to regulate her temperature? Is there any bedding material in the brooder that she could have choked on?
She didn't show signs of being hot or cold, and she could scoot around if she needed to. The "brooder" was just a temporary space to see if she would get better, so it was just a Tupperware with a towel in the bottom and a heat lamp overhead
Sounds like there probably wasn't anything you could do and there was just something wrong with the baby then. I've seen a lot of threads where people lost chicks to them getting too hot by using heat lamps so I thought I'd mention just in case.