Divine Smite allows my Paladin to expend a spell slot to deal extra damage after landing a melee attack. My character wields two swords. Since Divine Smite isn't a bonus action (because the rules don't say it is), I can use my bonus action to attack with my second sword while also using Divine Smite. So, providing both attacks land and I have two spell slots to use, can I use Divine Smite twice?
Yes, you can divine smite on any hit
The only limitation on how many times you can divine smite is your spell slots. But do note that with wielding two weapons, any bonus action smite casting will not be open to you unless you forgo an attack at some point to get it going.