Does having a high archery skill make enchanted bows last for longer before needing to be recharged? It's just that my nightingale bow appears to run out really frequently only in about 20 shots. My archery level is around 55 but my enchantment skill is incredibly low.

It ought to be related to your Destruction skill.

Those things are not connected in any way. If you enchant an item, the number of uses is based on the Soul Gem size you used to enchant it. Grand soul gem gives many more uses and a Petty.

With the Nightingale Bow it depends on your level when you got it. Look here; http://uesp.net/wiki/Skyrim:Nightingale_Bow#Nightingale_Bow

In past games, the higher your Enchant Skill, the less charge was used per cast. Not sure in this game.

Now it seems to be the magic school that covers the spell being done by the enchantment, and that just makes no sense to me.

Why?

IMO neither archery nor enchanting affects the charge. Once the item is enchanted, it has a fixed charge that should run out in roughly the same number of uses every time. If you acquired the Nightingale Bow very early (<18 level), the enchantment is quite weak and doesn't last for many hits. I don't remember the exact numbers, but it was along the lines of 20-something.

Now if you enchant your own weapons, the magnitude of effect is calculated from your Enchanting skill and number of uses is calculated from the soulgem you use. Again, your weapon skill has nothing to do with it. Certain Destruction perks will affect the magnitude of elemental damage, but not the charge of an already enchanted weapon.

It says on a loading screen that having a high level in the magic skill gives you more charges of an effect from that skill. UESP says "The total number of charges is increased by magicka cost reducing equipment and a high skill level in the respective magic school,"

Higher destruction skill = less charge used by items that create destruction spell effects. Same goes for conjuring (as with a Storm Atronach Staff) or restoration or whatever. Not sure on effects / items that combine two different schools.
This may only apply to staffs (which effectively cast spells) and not to enchanted weapons with similar effects.

The amount of charge used up is affected by your skill in the school of magic the enchantment came from. So, a bow enchanted with frenzy or calm will use less charge per shot if your illusion skill is high and a bow enchanted with fire/frost/shock damage will use less charge per shot if yoru destruction skill is high.

I thought it only worked when applying an enchantment, not when using weapons that are already enchanted. My bad. I really need to look into the crafting skills other than Alchemy