The short answer is, you failed to dodge tornadoes -> cannot DPS while being blown around -> lost too much DPS.

Tanks are pretty much locked into their Defensive stance in this phase, due to high incoming damage and the necessity of tank-swaps. So their DPS won't be too stellar.
Healers are pretty much locked into healing due to the high tank damage + unavoidable party damage. So you can't expect the world from them either.
You need DPS who can dodge the tornadoes consistently while still keeping up their rotations.

In a pinch you could use Melee LB to help kill one. Also if you're crafty and can work out the tank facing him at a 90 degree angle (careful where you stand, it cleaves), it's possible to Ranged LB both snakes at once. It's not recommended to use Caster LB since if they get too close their damage shoots through the roof (as if they didn't hit hard enough already).

If you're playing with people you know, force them to buy and use food. Or even buy it yourself and give them out before queuing. If you use a stat pot on the Chitlin, it will not be available for the snakes. But if you can down the Chitlin without using a stat pot, perhaps you can use a stat pot for the snakes to help out.

It should be obvious that skill > gear. Quite a number of people have cleared this encounter without even having ilvl170.