Oh dear. I agree that the death penalty should be carried out much faster than it currently is (if it is going to be kept), and yet even I can't find much to say that's good about your essay.
First, you don't deal with the issue of potential miscarriages of justice if someone should be wrongfully convicted. The reason the U.S. has such long delays between the sentence being handed out and the sentence being executed, as it were, is that it provides plenty of time for new evidence to be dug up, appeals to be filed, etc. You can't just ignore this if you hope to be convincing.
Second, you seem to believe that the alternative to the death penalty is the release of the condemned back into society:
Imagine the liberation of dangerous and homicidal prisoners into your neighborhood
. As in fact convicted murderers can be kept in jail for life, this is not something that is actually likely to happen.
Third, while your argument that life in prison could just as easily be considered as "cruel and inhumane" as the death penalty could potentially be strong, your method of phrasing weakens it, as you approach it from the point of view of the inmates. I assure you, most people, given a choice, take life over death, even if it is life in prison.
Fourth, none of this addresses the actual prompt, which is whether or not the death penalty is "fair." To do this, you would have to look at whether it is just to take the life of someone who has committed murder (Hint: the answer is pretty clearly yes, under just about any definition of justice that hasn't been tortured by a philosopher specifically trying to defend the abolition of capital punishment). You would also have to look at whether it would be just to execute someone for a crime that did not involve killing (treason, rape). This becomes a harder argument to make. Finally, you might also look at the way the death penalty is actually applied in your country. Some argue that the racial imbalances in the U.S. in the application of the death penalty, for instance, make the death penalty unjust. The argument isn't a very good one, but you probably should deal with it. Also, you will still need to look at the issue of the possibility that the wrongfully accused might be sentenced to death.