2 diclaimers here. The first ring is the AP and armor one, the second one is str8 AP. Next disclaimer is yes I know AS Irelia is more viable, I player her that way mostly.

Summoner skills:
exhaust
Ghost

Others
teleport
ignite
flash

your choice if personal preference though. I usually do teleport and exhaust/ghost

This build is all about AP Nuke and sheen/lichbane. After activating your W and after every Q and E your lichbane will proc. On your Are it procs after each blade thrown as long as you space out the Throws.

So lets take some numbers. Lets say you have LB rylais and a ring. That's 80+80+100 so 260. So your 1 does 560 base and your next attack does damage plus 260. Okay now activate your W. Another proc that's about 360 and your melee hits harder. Then activate Q that's 100+100 and another 360. So in 8

seconds you've done 560+360+360+200+360 and your melee attacks. That's a perfect storm tho. Using abilities and owning with LB procs.

what's nice is your ultimate. It does 190+.4x280. So about 190+110 per blade per champion. Although at level 16 you should have more then 3 AP pieces. Full build gives 80+80+100+70+130x1.3. About 580. So your Q is doing nearly 900 and your ultimate is doing about 190+230 so roughly 420 per blade times 4 so around 1700 which does not have DR's. Ita just a nice build. Obviously if they stack mr MAKE VOID STAFF OVER ABYSS.

Runes I chose:
MP Marks
MP glyphs
CDR Seals
HP Quints

If your confident you an go MP qunits. AP Nuke Irelia does well in most situations and the build can vary. She can be viable with defensive items. Also on a side note nashlor is a very good item I'd get it over the second AP ring most times. AS is good with your W AP and CDR is gg. A very good second item rather then rylai's. I get rylais 3rd due to hp and slow on Q and ult. There is in truth alot of ways to play her tho.