Is my wireless card broken?

The problem: When I load up my computer, the wireless internet works fine for about 5 or 10 minutes and then disables itself all of a sudden. The WiFi light on my laptop just goes off and I'm unable to enable it in my network connections. When I restart the computer the wireless internet is again working fine but will again disable itself after about 5 or 10 minutes. The internet works fine when I plug the computer into my modem though.

The question: Is my wireless card banjaxed? If it is, how much would it cost to replace it?

Sounds like it. Is it a card or onboard? If it's a card, take it out and put it into another machine; if the same thing happens, it's the card. If it doesn't try reinstalling your drivers. Or vice versa, if you don't have another machine to hand.

Unfortunately it's built into the computer. Would opening it up and changing it myself be a big deal or would I be better leaving it into a PC repair shop?

Built-in doesn't necessarily mean it can't be replaced. Is it a desktop or a laptop? How new is it? Modern (<2-3 years) laptops and some desktops have the wireless built right onto the motherboard, in which case you'll probably need to buy a plugin card. Replacing a motherboard generally isn't cost-effective.

It's a Dell laptop that I got three years ago this month. I found this page on dell.ie that explains the exact problem I'm having. Hopefully once I've downloaded and installed this driver it'll do the job. :)

Excellent!