Please don't give up on this dog, none of this was her fault at all. Sure, it is distressing for your neighbours and you will have to be more mindful in future but she doesn't deserve to be rehomed because of it. How will that even help change the behaviour? There will always be cats around no matter who she lives with.
I actually had a horrendous experience like this not too long ago so I understand your guilt/panic. Both my greyhound and terrier got hold of a cat in the yard. I literally could do nothing to control the situation as it flew wildly hitting off the walls etc trying to escape but kept getting grabbed
Eventually it managed to get away but I was terrified that it was bleeding to death somewhere in an alley or something. I called the RSPCA and went out searching (this was late at night) but never found it
Both my dogs had injuries and Hooch had a severe infection in her eye. Count yourself lucky that your dog is ok 
I know this situation is horrible but your dog just got over excited. Accidents happen
I actually had a horrendous experience like this not too long ago so I understand your guilt/panic. Both my greyhound and terrier got hold of a cat in the yard. I literally could do nothing to control the situation as it flew wildly hitting off the walls etc trying to escape but kept getting grabbed
I know this situation is horrible but your dog just got over excited. Accidents happen