I don't think so, I think the spell works only with personally cast spells, not enchanted weapons.
The fire/frost/shock damage buffs in the destruction tree effect their respective damage on a sword enchant (have to have the perks before you enchant it). I would think, and read (I really wish I could remember the source) that silent spell would work the same way.
But, to get back on topic, perks should be applicable to weapons, be it sword or staff.
Also, the twin soul perk works on summoning staffs (or staves as the debate goes). Yet another perk that effects them.