What should I do if my laptop’s power button wire is cut?

You might want to consider just replacing the wire entirely. You can find replacement parts online, like on eBay. There's even a YouTube video that shows how to do it, starting at about 2:25. That could save you some hassle in the long run!

Alternatively, you can try re-wiring the power button. It's usually a temporary fix, but it should work until you can replace that wire or solder it properly.