By the definition of "broken"? What is this definition?

Broken = not working. A system where an innocent person is convicted to death, is not working correctly. Broken.

Personally I think a so called perfect justice system is an illusion as long as people have to work with it and it's hard to get to the real truth of the matter.
People [who work for justice] make mistakes, they can be influenced by media, political climate etc. Proof can point in one way, yet with additional evidence can point into a completely other direction.
I don't think the death sentence should be abolished just because the system is not working like it should. Any system has flaws as long as people have to work with it, and proof can be hard to establish.
And that is the reason the death sentence should be abolished.

There are uses of the word like that but I wanted to know what Neil Blanchard meant by the word because he was making particular assertions using it.

Your argument depends on the word you have put in bold, the word "that". If you could say what that refers to it would make things clearer. Is it the difficulty of proof (proof beyond reasonable doubt in the minds of the jury, proof beyond reasonable doubt in itself, or some other notion?)?

"That" refers to:

To elaborate, it wouldn't be the first time that 5 years after a conviction, additional evidence comes at hand, and the accused on death row has to be released because he or she is innocent.

And it wouldn't be the first time that a witness was in error.
People are people. Perception is a weird thing.

"Beyond a reasonable doubt" is not safe enough for me when it comes to a life of a person. Beyond reasonable doubt, there is still a calculated margin for error that the conviction is false.
I wouldn't want for anyone to be in the margin of error.

"12 angry men" is a great movie and shows what I mean, to some extent.

Aside from that, I find the death sentence barbaric. It doesn't belong in a civilised society. People who cringe when they hear they chop off hands of thiefs in other countries, should cringe by the words death sentence as well.

The two parts are related: doctors are well qualified (medically) to properly administer the injection -- but they are barred from taking part by their oath to do no harm. The poor persons who's job it is to execute the person are not medically trained (in all cases, anyway), and they obviously, can make mistakes.

Also, the three drug cocktail that Florida uses, along with most other States that still execute people, is simply not always effective, and (apparently) it can be done incorrectly. And, it might be causing a lot of pain before killing the person.