The USB port on the Brutes does not transmit audio. You will need some other method of connecting your Brute to your PC. Needing a sound card is a bit bigger question. What you really need is an audio interface. Something that will physically allow you to connect the audio out from the Brute to your computer. Most (though not all) interfaces will act as a sound card. Almost all sound cards have an audio input but they are typically 1/8th inch and offer very little flexibility.
Your audio interface will become a very fundamental pat of your recording 'system' so this is no place to skimp. However, there are many very affordable interfaces on the market now and for as little as $40 or $50 you should be able to find something that will work for your needs. Of course, you need to figure out what those needs are. Try to think ahead about what you might want to record in the future. The number of channels is a hugely important requirement. If you ONLY want to record your Brute, nearly anything will do. If later, you decide you want to record a whole room with many mics etc, you will need more inputs.
One feature I would be sure to get with your interface though is ASIO. It's the latest protocol for passing sound back and forth between a computer and an interface (assuming you're in Windows of course) Previous protocols (WDM, etc) work just fine but don't offer what ASIO does. Fortunately, even the least expensive interfaces now seem to support ASIO.
The EIE you mention is an excellent choice. I have one myself. The fact that it has 4 usb ports is particularly useful to me.
Best of luck,