You can not stream the audio from Xbox 360. At least, not without a lot of mucking around. Even with a headset, you will generally not hear the regular game sound, as it was designed for player communication. There should be an easy fix for routing the sound into a speaker.
From what I gather, you are using HDMI. HDMI binds audio and video into a single connection, compared to AV, which has a separate plug for video, left sound, and right sound. Since AV sounds like it might be an easier fix of the audio problem, I'll tell you what to do for both.
You have a couple of audio connections. I'll give you an overview, but remember, when it comes to connection and adapting it almost always follows the rule of "if it connects together it will work".
- 3.5mm is the connection you would typically use for earphones. It is called 3.5mm, as that is the width of the actual plug. Don't mistake it for 2.5mm, which follows the same rules. It is the standard connection for your generic computer speakers. If you use this connection, ensure it has at least 2 rings, as the rings represent channels. A single channel 3.5mm will only give you one side of the audio.
- RCA is the connection you use on the TV, pre-HDMI. The red and white carry sound. This connection is a lot more common, these days, on speakers. It can easily be adapted to 3.5mm. The Xbox 360 uses RCA for its standard-definition output. Ensure you keep the colours matched, when using RCA, as the colours represent channels. You won't miss out on a side, but you might get confused when something moves to the left, and you hear it from your right speaker.
- Toslink is an optical connection standard to high-definition. If you have a newer sound system, it will likely have this connection. So does the Xbox 360. You can get these cables pretty cheap, if you go to the right place.
You actually have two methods to keep your HDMI, and corresponding video quality.
If you happen to have a decent sound system, it will likely have digital connections. You can find the cables (Toslink) pretty cheap, and the Xbox 360 has a connection on the back. Regardless of AV or HDMI, it is plug in and play. You can also find converters, to use 3.5mm or RCA, but this may be out of your budget. Furthermore, if you are going for the adapter, it may be better to use the HDMI audio converter. As soon as you adapt the Toslink into another connection, you will likely lose all associated sound quality, as the other connection will "bottleneck" your signal.
Using a HDMI audio converter
You can buy a HDMI audio converter, which fits between the Xbox 360, and the HDMI cable. This effectively gives you a second auxiliary output for the sound, where you can use a standard 3.5mm or RCA cable.
If you can get away with running the RCA cables, you have another option. Given that the sound cables run separate, you will find the two sound RCA plugs fit directly into most speakers. Alternatively, you can get an RCA to 3.5mm adapter, to connect it to regular computer speakers. I have not seen as much for converting DVI to RCA, so you may have an issue, on the video side.
If you look around, on eBay, the parts can be very cheap. I work in electronics retail, and it surprises me how much markup there is on this stuff. People will always pay $50 for a $5 cable. They don't know any better. But with audio, it really is mostly "plug and play". If you have any electronic repairers, check with them. Electronic retailers? Go in and ask for their most unsellable product. Unless the cable is physically damaged, there will be nothing wrong with it, but it is next to impossible to sell a cable that somebody has torn out of the box. They would also likely have "shop spares", previously packed with devices that may have been written off, or otherwise rendered unsellable.
You can find a guide to connecting external speakers on an Xbox 360 here, at Microsofts official Xbox support site.