Hi, I was just wondering whether I should level up my destruction magic and invest in the right perks so that my relevant enchanted weapons last longer before I need to recharge them? I want to know before i go off wasting time levelling up this skill, because I'm finding that my blades run out of "fuel" after about 6-7 swings.

Note: This is on my thief character who's level 40 and has rarely ever used destruction magic in his life (currently skill level at 21).

The duration they last is down to your enchantment level when you create them and the actual value you choose while doing so, you can alter a slider selecting between power and charge ratio.
Destruction magic can be levelled using staves though, especially using ones taken from Dragonpriests.

I didn't create my weapons, I found them.

Neither Destruction nor Enchanting has any effect on found weapons that already have an enchanted effect. The duration of the effect is set when YOU craft the enchantment in the weapon. That level is based on your Enchantment skill. Destruction affects the use of staves as in a fire mage and fire staff.