I don't think I saw Veigar or LB. Those two are the AP burst mages-LB is early game domoniation, and late game assasin. Veigar is ok early, to farm up his AP, and late game he just booms.
Ahri, Kat, Akali, Diana, Karthus, Gragas, Kassadin, Mord, Zig, Fizz, Ori, Morg, Ryze, Zyra, Swain, Annie, Syndra, Malz, Anivia (God that passive GA), Thresh, Eve, Cass, Vlad (But top is better, because most solo tops are tanky melee, but his tanky AP with super sustain, though damage is not as amazing as others), Zed, Brand.
All i can think of right now for mid champs you should try. I don't own all of them, but 95% of them, though I obviously faced them all and used them for free before.
Akalia- Energy base, u can trig both her innate with mastery and runes, free spell vamp lvl1, has stealth and can be super hard to kill at lvl4 while she pwns lane most of time at lvl6. Has triple blink.
Diana- Tanky for a AP mid. No cd ulty for blink and has a refresh shield to win unexpected fights. A gap closing slow makes it impossible to escape alone, unless for flash or summoner spells. Q is odd but the damage is decent.
Brand- Just beast passive, high damage out put and strong CC (takes skills to make the aim, but really scary when pull off). Farm power is great and easy to harass. 3 AoE skills, while ulty is just ridiculously. The downside is the squishiness but he has nice cast range. His also really mana hungry.
Cass- Only downside is the need of mana. 2 spammable skill during lanning, while increase movement will allow her to easily trigger her E. W is quite useless until u have ult, but it can zone and lock down path ways, help team fight, or escape. Ulty is not the best in the game, but pulling of the immobile is deadly as hell. While escape, if have rylais, u can keep spamming Q till u think u can win a fight, just E a lot of times
Eve: Last but no least, I don't play Eve that much as a Mid, though I do sometimes. Stealth is key and piss many ppl off. She does fall off late game, you are warned.Though her early power in limiting the enemy mid is strong, unless u fight someone like Zyra or Heimer who spammes unit or a mage counter pick like LB or Kass. Her Q is piss off for farm, chase damage. E is just strong, W solves the limited stealth and chase. Mana is np because she gains mana regen on innate when stealth.
Fizz, Ahri: Oops XD, I have more fav, those 2 are super mobile (Like LB but LB is way too squishy), Ahri has triple flash while Fizz have untargetable and dash. They are neutral in all aspect, because they can be tanky, bursty, mobile to engage or retreat, but not amazing in any aspect. Ahri has a cc but not reliable, fizz has slow which is not reliable too. Ahri has true damage Q, which is good, W for close range, also nice, and a spell vamp innate, though she's mana hungry early. Fizz is prob better, but I just prefer Ahri, because Fizz is simply harder to play.
If ur beignner, play OP like LB, Brand
If ur experience, play Kat, Karthus
If ur long term active addicted fan of LOL, Play Ahri, Diana, Akali, or if u want to challenge, play Talon, Zed (all the Melee base AD champs that are viable for Mid)