Sprint Samsung Galaxy S3 Dropped phone now it won't power on or charge
That's a very weird situation. You would assume it's a hardware problem... which it probably is, but it's possible the filesystem could've corrupted when the phone lost power after hitting the ground.

I would try inspecting/replacing hardware first (only if you're experienced), just reseating all the ribbon cables, maybe lightly clean the contacts with isopropyl alcohol to remove oxidation (again, only if you're experienced).

The very next thing I would do, is download the "stock SPH-L710 firmware/ROM image" file off XDA-Developers and use the instructions/downloads provided there to flash the image in download mode via Samsung's Odin program. They make it sound like it will wipe all your data, but from my experience (I have a Galaxy Note 2 -Verizon which I bricked), it will not wipe your data as long as you flash the correct image, same Android version (likely 4.1.2 or 4.3). If you don't know, and you know you have updated the device regularly, I would try to find the latest version available.

If you want more details on either of the solutions, I'll edit my answer to provide more details/articles.

Hope this helps,