Why is the melting point of hydrogen fluoride lower than that of ammonia? The melting point of hydrogen fluoride is -83.6°C, as compared to that of ammonia, which is -77.73°C. How does this make any sense?

Both hydrogen fluoride and ammonia show hydrogen bonding, which "elevates" the boiling points of these two compounds when compared to the rest of the pnictogen and halogen hydrides. However, I was under the impression that hydrogen bonding increased in strength the more elctronegative atom the hydrogen atom is bound to, explaining why the boiling point of ammonia is significantly lesser than that of hydrogen fluoride (-33.34°C and 19.5°C respectively). 

However, this makes no sense when looking at melting points of the two elements.
The additional energy involved here is the stabilization energy, a form of potential energy that is stored in the more stable arrangement of a solid compared to the liquid form. This energy is determined by many factors:

Size and shape of the individual particles.
How they can 'pack' into a solid arrangement.
etc...

SO, there is clearly something different about how Ammonia is able to arrange in solid form that makes the solid stable for a greater range of energy added.
When we look at a phase change diagram the slope and horizontal component of the line determine total energy needed for that phase change. This will be different for each substance considered, and this is where your discrepancy comes from.