Dropped iPhone 4S, stuck in Recovery

So I tried putting my iPhone 4S into my pocket but missed and it fell directly on the corner from about 3 feet and hit the floor (hardwood, a basketball court to be exact). Surprisingly, the screen or back did not shatter, but it no longer powers on. It also got really hot until the battery died.

I also checked all of the connections and they seem to be fine, along with unplugging the battery and plugging it back in. When I plug it into my MacBook, it immediately goes into recovery mode according to iTunes. The screen stays black, but is detected in recovery mode (maybe DFU?) by iTunes. I tried restoring, but it stays at "waiting for iPhone".

I feel like this is a logic board issue, but I can be completely wrong. This is just a spare phone so it wouldn't be a major loss, but would be nice for it to work again.

It is worth noting that when I plugged a different battery in, it got hot again just like before. Also, there is no liquid damage, and I am unsure if it is the screen that is causing this, but that shouldn't be throwing it into recovery.

If there is anything that I am missing or any information that you need to help, please let me know. I really want this iPhone to work again.