Get Ap. Read some singed guides and builds theres tons of em.
www.mobafire.com

I believe late game he is a great initiator on 5v5 team fights. If you bulit a good armor/Mresist and hit the ult, send a glue on the back, use ghost and just dive into the enemy team looking for the squishy carry and fling it on the mid of your team to be masacred. It immediatly turns it into 4v5 and you can for sure get out alive cos endgame Singed is just immortal (if you bult it as a strong tank). That's how i play it at least.

For a massive damage go with an AP build, there's many on the guides, but i'm not an expert there cos i believe it's true potential is as a heavy tank.

Endgame if you build
ROA
Merc Treds
Soul Stealer
GA
Death Cap
last item get
Some magic pen or Defensive depending on other team.

check my profile if you want.

endgame you will have damage and tankyness

Ah, Singed. No matter how many champs I fall in love with, Singed will always be my number 1. Check my profile to see I play ranked almost exclusively with Singed.

You asked about Singed's damage output late game, and I'm afraid that just isn't his late game role. His late game strength is, at it was already said above, the ability to dive into 5 enemy champs, grab the enemy champ who relies on their distance from the fight the most (ashe, twitch, kog'maw, any high damage ranged carry) and toss them where they do not want to be. Everything else is self explanatory, slowing enemies, gassing them, taking a ton of damage so everyone else doesn't have to, etc.

That isn't to say you can't stack damage if you want, but please believe me when I say that you are most effective, even if your goal is to kill, when you can take a severe beating. To this end I recommend the following to enhance your damage while still performing your role:

Rod of Ages
Rylai's Crystal Scepter
Abyssal Scepter

All of these items provide an increase to your ability damage while enhancing your survivability. I personally never get Abyssal but perhaps it will suit your aims. Good luck.

FON, Shoes, Abyssal, Rod of Ages, Rylai's, Hourglass
Give or take a banshees or thornmail

Gives a good mix of both defense/mresist/HP and AP. 3000+Hp, 150+ on both defense/Mresist, and 250+ AP (cant give you exact numbers). Add in some defense/MR runes and masteries, and you'll be able to tank a lot while dealing a lot of poison damage. Only thing to replace is in certain games where you need Banshees, Thornmail, or GA.

a rod of ages and rylias is usually enough damage to wreck squishies endgame, but as for tanks just throw them into you teamates and let them handle it