I don't really know what your setup or feeding is like, but young geckos can be pretty fragile, so the combination of having a recent tail drop, a lot of handling, high heat, etc. could have just put too much stress on its system. Babies really need to be, well, babied.
I'll post a link to one of my favorite care sheets that sort of outlines the simplest possible enclosure and ideal conditions:
If the room is that hot regularly, I'm not certain how much luck you'll have without providing the gecko a way to cool down. Having plenty of hydration and a humid hide available can help, but leopard geckos like to hide from the heat and don't really thrive in it.