It could be the logic board, but your description sounds like the power from the charger either isn't reaching the battery, or the battery can't accept the power. In both cases, the charge applied through the Lightning connector has to go somewhere, so it's dissipated as heat.
I would check to see that the drop didn't knock one or more connections loose. The connections that I think might result in these symptoms are:
- the connection between the Lightning Connector module (also includes the headphone jack) and the logic board
- the connection between the battery and the logic board
- the connection between the power button/audio control cable and the logic board
iPhone 5 Lightning Connector and Headphone Jack Replacement
iPhone 5 Dock Connector and Headphone Jack
iPhone 5 Battery Replacement
iPhone 5 Replacement Battery
iPhone 5 Audio Control and Power Button Cable Replacement
iPhone 5 Audio Control and Power Button Cable
Check these three connections, and replace any cables/modules that appear to be damaged. If a cable looks torn, or a connector or board is cracked, that's a strong candidate for replacement. Since you're inside the phone anyway, this is a good opportunity to swap out the original battery if you've been getting reduced charge life.