Yeah, he fumbles a bit at a couple of points there. Speed isn't everything!
But the vid does at least give you an overview of what it may involve, and as you can see, it's not horribly daunting as long as you keep organized and don't mislay screws etc.
You need to take basic antistatic precautions any time you're working inside a computer. You'll at least need some decent precision screwdrivers, but they're not expensive, and can be handy for other things. It can be a good idea to keep a camera handy and take photos as you go along so that you can see how things originally fit together. You also need to be gentle with any connectors that you need to disengage, as sometimes they're quite fragile (you'll maybe notice that he has trouble re-connecting the power line to the motherboard - ends up he had it upside down!).
I'd prefer to have a proper service manual to hand for a task like this, and it would no doubt help you (though apparently Sony doesn't publicly release these manuals, unlike Dell, for instance).
Without knowing exactly what model your VAIO is, there's only so much I can chase up on Google, but there's peer support forums out there. Here's a thread on one (which I found in just a couple of minutes) all about disassembling VAIOs, where a helpful professional tech has made a number of his own guides available, including "Sony Vaio VGN-S360P. Replacing broken power jack."
The thread's here: http://forum.notebookreview.com/sony/317635-sony-vaio-disassembly-guides-instructions.html
Direct link to the guide: http://www.insidemylaptop.com/take-apart-sony-vaio-vgn-s360p-replace-broken-power-jack/
However, note that this VAIO is a different model to the one in the video above - like Dells, for this one you have to go in through the keyboard rather than from underneath, so disassembly is a bit different, and also note that this one does have a soldered power connector. Let's hope yours is a different model and doesn't need soldering, but if it does, the tech describes how to do it.
His range of other guides for Sonys (if he doesn't tell you specifically how to replace the power connector for yours, you may at least get instructions on disassembly) is here: http://www.insidemylaptop.com/category/7-sony-laptops/
If that's not your model, it's worth reading through that whole thread in case yours is posted later. (There's a chance that the guide for one model will be OK for others, but I've no idea about that.) As you'll see, folks who're doing it are asking questions as they go along, and some of those may be of help to you, too.
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