So...for clarification: you want the switch to operate the (new) lights, instead of the half-outlet? Or you want the switch to operate both the lights AND the half-outlet?
How many lines of romex are in the outlet box? As far as I can tell:
Switchbox: only one romex-- hot in, switched out, and ground. Do you know whether black or white is the switched wire?
Outletbox: three romex? one to/from the switchbox, one from the breaker, and one...to your lights? I'm trying to make sense of the spaghetti in the second picture.
It seems to me that you should ultimately have a romex line coming IN to your outlet box (always hot black + ground + neutral white); a romex line going from your outletbox to your switchbox; and a romex line going from your outletbox to your new lights:
Assuming you want the bottom half of your outlet to remain switched. Make sure you break out the tab between the two hot (brass-coloured) screws so the two halves of the outlet can be powered separately, or you'll end up with bad juju.
Romex1[powerwire]-- Black to a Hotwirenut, White to a Whitewirenut, Ground to a Groundwirenut.
Romex2[switchwire]-- Black to Hotwirenut, White (put black tape on it) to tapedWhitewirenut, Ground to Groundwirenut
Romex3[lightwire]-- Black to tapedWhitewirenut, White to Whitewirenut, Ground to Groundwirenut
Jumper1-- Black from Hotwirenut to non-switched (top) hot side of outlet
Jumper2-- White from Whitewirenut to neutral side of outlet
Jumper3-- Ground from Groundwirenut to groundscrew on outlet
Jumper4-- TapedWhite (or black) from tapedWhite wirenut to switched (bottom) hot side of outlet.
You may be able to make it less complicated in the box than I've written it out here, but this should get you started.
Disclaimer: I AM NOT AN ELECTRICIAN. I am a handyman, and I've done a bunch of stuff, but I AM NOT AN ELECTRICIAN. Please don't be stupid and kill yourself or burn your house down. The internet is your friend. There are a lot of very good wiring diagrams written by certified electricians available with a quick google search. If you don't think you understand how wiring is supposed to work, please take the time to look it up and make sure you understand. No-one wants to be electrocuted.