Welcome to Android Central! There are two main things to suspect:
1. There may be a corrupt file on the SD card. This can cause the system's Media Scanner to hang up while it's trying to scan that file, causing it to eat up battery. Unmount the card (in Settings>Storage), remove it, and insert it into your Windows PC (using a fullsize SD card adapter, very cheap to buy if you don't have one). Then run chkdsk to see if there are any bad sectors or corrupt files:
http://forums.androidcentral.com/amb...d-sd-card.html. Also see how the phone runs without the SD card--if the problems resolve, then the SD card was the problem.
2. If the SD seems ok, then try a new battery. Batteries can fail suddenly, and often exhibit signs like quick discharge or abnormal heating up.
3. You could also try booting into Recovery and wiping the cache partition, which doesn't erase any personal data:
Reset: Samsung Galaxy S 4 | T-Mobile Support