Ok, your problem imho here is a disconnect between the needs of an individual and the needs of a society, and understanding that the latter inevitably includes somewhat overlooking the needs of the former.
As a society, we need certain laws and practices enshrined as goods in and of themselves in order to prevent descent into a Hobbesian rule of all: a state of anarchy and emotionalized revenge that we know would in the end only increase suffering, astronomically. But this is sort of a meta-good, rather than a direct one: it lives or dies as a rule for everyone, the shape of an entire society, rather than as something that necessarily makes sense in every individual situation.
So in this case we have a case in which a man has done something abhorrent and obviously is, at least at the moment he said it, displaying little remorse or at least some bitterness and nastiness for everyone involved in his punishment and suffering (which, if we are fair, cannot be discounted either: being in jail is a miserable thing, regardless of what you did). The poor mother interprets this as having no remorse at all, and kills him in order to assuage both her conscience (instinctively fearing that such a beast may rape again) and her sense that evil done should show remorse, not glee.
The man dies, and this is, in itself, probably an injustice: for whatever flaws he had for not being anything but grovellingly sorry to the mother for the sufferring he caused (a state which may be caused by his own evil sociopathy, but just as likely could be caused by his own pain and resentment at having been made to suffer so much and for so long), immediate, in the moment cessation of life is probably unjust.
But the problem is that we are not beings who can viscerally accept being a part of a society of laws. We have moral feelings and feelings of revenge and protection that are primal, and a system of laws is a higher order thing that only really makes sense in the long term, in the big picture. So we are at war here, in our natures, between feelings of payback and abhorrence and feelings of what is good in the long run.
The refined idea here is that I don’t think you SHOULD, ever, expect to feel okay about either side of this controversy. If you are a good moral human being, then this situation SHOULD make you feel uncomfortable, and unfortunately, is it one that probably SHOULD leave you with a sense of upset that has no true resolution. Because it IS an uneasy bargain that we’ve made, as members of a society but also as emotional beings. These two things cannot be cleanly reconciled, and that is going to cause you, a moral caring being, unresolvable stress. We’re inevitably talking a trade-off here, and either prong of the dilemma is going to hurt someone. And as a being of empathy, that is not ok with you: that is not “Pareto optimal” or ultimately acceptable for someone that wishes for a perfect world.