Before I say anything else, let me start by saying that I understand your frustration and yes, even your anger. If my kid was grounded for three months for posting a similar rant on her Facebook profile and s/he did it again, I'd be upset too. You are well within your rights to discipline your child. Parents should do that so their children grow up to be responsible adults.
That said, I question your methods, sir.
As I was watching your video, it didn't sit right with me. Something was bothering me. Like I said, I understand your frustration. I can sympathize with it. But as I watched this, I couldn't help but feel a bit uneasy about the whole situation.
And it's not about the gun. No, really, it's not, although I do wonder whether or not using a weapon of any kind to discipline your child is the best idea.
Instead, it's about the overarching methodology you've chosen.
Let me break down the situation as I'm seeing it:
A daughter is unjustifiably upset by what her parents have done, and so she posts what amounts to a temper tantrum that's visible only to her friends (or at least, it was supposed to be).
Then a father is justifiably upset by what his daughter has done, and so he posts what amounts to a temper tantrum that's visible to the entire world.
Do you see the difference between those two statements? There are two: whether or not the person was justified in their anger and the size of their audience.
Again, I'm not condoning your daughter's behavior. It was reprehensible. Referring to a friend of the family as "the cleaning lady" is not cool. Whining about a short list of chores in a profanity-laced post is not, in any way, okay. And you're right, she does seem to have a gratitude problem.
But here's the ironic thing: you're upset about your daughter airing the family laundry in a semi-public forum and so, to discipline her, you air even more of the family's dirty laundry in an even more public forum.
Sir, you are supposed to be setting the example in your family. Right now, the example you're presenting is that it really is okay to post a public rant. And I'm pretty sure that's not the lesson you're trying to teach.
If you're a God-fearing man, you're probably thinking of what we Lutherans call the Fourth Commandment right now, Honor your father and your mother. And that's certainly true. But after watching your video, I couldn't help but think of a different part of Scripture, namely Ephesians 6:4. "Fathers, do not exasperate your children; instead, bring them up in the training and instruction of the Lord"
Might I suggest an alternate approach? Instead of filming a one-sided video rant and posting it for all the world to see, maybe actually sit down and talk with your daughter and explain why you're so upset. Discipline her by all means. Take away her laptop to remove the temptation. But do so in a loving, private manner.
Oh, and if I may be so bold, I'm still not sold on using the gun. Maybe, instead of destroying the laptop, you could have deleted her personal data and given it to the woman who has been helping out around your house. I understand she's down on her luck and might be able to use it. Just a thought.
A concerned fellow parent