Listen, it's NOT your fault -- **** happens, and it's a learning experience. In my house, our cats are completely disinterested in the chicks, so we don't think to take any extra precautions with them. If something did happen, I wouldn't blame myself; sometimes, an animal's most ancient insticts supercedes it's domestication, and bad things happen. We had an incident where our dog (who is almost always crated whenever I have to move/remove/clean out the brooder) managed to unlock his crate, sneak out without me realizing, and as I was removing a chick from the brooder, actually went to EAT the chick. Thank God my husband caught him as he was doing it and pulled it off, but it's an example that proves no matter how careful you are, no matter what precautions you take, something unexpected can always happen with animals, including cats and dogs. Anyone blaming you for the accident is clearly fortunate enough to have never experienced something like this.. but that isn't an excuse to be ignorant.
Anyway, for the future, this is what we do: we use a BIG (I want to say 50 gallon?) Rubbermaid storage container, sans the lid. We then put the Rubbermaid container inside of an old dog crate (for large-sized dogs -- it actually used to belong to our rottie, who passed away 2 years ago), which was initially to keep the chicks from hopping out and to prevent our quail chicks from flying out. It also works amazingly well at keeping out cats, dogs, our 4-year-old hands and anything/anyone else who shouldn't be in the brooder. We hang the heating lamp off the top of the crate with a large clip. This is what our brooder set-up currently looks like (ignore all that mess on top, it's usually clear on top!):
Our isolation brooder is a similar set-up: it's a smaller Rubbermaid container (around 20-25 gallons), and locked up inside of a medium-sized dog crate when in use. You could also use a large cat carrier or a similar cage with a Rubbermaid container for that purpose.
And don't get rid of the cat -- it's not her fault, either! It's her instinct to chase birds, so if she sees a baby chick, she's not going to understand that they're off limits. I probably wouldn't try to re-introduce her to chickens, either, even older ones; if she attacked once, she's probably always going to want to attack again, and nothing you can do can remove that desire. You may be able to teach her not to do it (and older chickens, at least in our house, will certainly put her in her place if she tries), but you can never be 100% sure that temptation won't overcome and cause her to attack again. The best thing to do is to keep the brooder locked up inside of a crate where she can't reach it, and when you need to remove the chicks, lock the cat up in your bedroom until everyone is back safely inside the brooder. Once the chickens are outside and free ranging (if that's what you plan), they should be old enough to fight back, assuming you allow your cat outdoors. If you can, though, you can keep the chickens in an area where the cat can't reach, maybe in an enclosed run or something. But worry about that when you get that far, because that'll likely be a lot of trial-and-error if your cat is anything like ours!
Good luck, I hope you try again, and I'm really sorry this happened. But try and take it as a learning experience; I've lost quite a few chicks/chickens, some to illness/injury and others to hawks/predators, and it's more upsetting each time, but I try and take each loss as an opportunity to correct what I'm doing wrong and/or improve the way I care for and protect my chickens. That's all you can do, really, and hope for the best :)