I'm curious as to what the minimum internet speed is to allow access to Xbox Live.
What minimum speed does it require?
It's really difficult to say as it depends on so many other different factors as well. At first glance your download speed look acceptable, but upload speed is pretty low and your ping is fairly high as well which will both have an impact. There's also the issue of contention, which may see your broadband speeds rise\fall depending on how many other households are using the line at that particular time.
Then there's the type of game\service you want to use. Some games require more bandwidth than others, for example a first person shooter game will require a lot more than say, a turn based game like Uno. Streaming films also requires a lot, depending on the quality.
So taking into account these factors here's my recommendation:
Heavy bandwidth games (COD, Halo): Most likely have lag, and be hard to play online
although don't take this as definitive, you can hook up to xbox live for free and you usually get a months free gold membership too so you can test out some online gaming before you commit to a paid membership, and I'd advise you ultimately to do this.
If your ping stays the same, fast-paced games like first-person shooters will be miserable. Ping is the response time (in milliseconds), so your ping stands at 0.879 seconds. Think about trying to shoot someone when you have a delay of almost a second before the bullet fires. About like driving when you are drunk. w69rdy covered pretty much everything else, just wanted to make sure you knew exactly how the ping related.
As far as I'm aware there isn't a minimum 'speed'.
Rather Xbox Live requires an MTU (maximum transmission upload) of 1364, most Ethernet routers should have this set to ~1500 so you shouldn't have to worry about this.
It is entirely possible to access the service through a 56k dial-up connection and there are several tutorials on the web demonstrating this. This is extremely useful in countries where broadband is either too expensive or just unavailable / unreliable.
However this will obviously create problems, you won't be able to play multi-player, at least not without ruining the experience for you and your fellow players. Getting a gold subscription wouldn't be worth it but if you just want to download games from XBL and use some of the non-bandwidth intensive parts of the service its fine.
As far as the minimum speed required to play consistently smooth, stream video and enjoy the 'full' xbox live experience, that is much more complicated and actually depends less on the 'speed' quoted by the provider and more on other things. I would say the most important factor here is how close you are to your local exchange and also how densely populated the area is and the time of day. If you live miles away in a densely populated area you will have a worse experience.
Multi-player will also depend a lot on your global location, for example if a game you play only has servers in Europe and you're an American you're not going to have the best experience, and vice-versa, this is an over simplification though.
As recommended your best bet is to simply get a free gold month and test it out for yourself, most multi-player games come with at least a few days trial. If it is poor then remember it may be because you've been connected to people half way across the world, try a few games, try a different time of day (if its a busy period like 6-10pm on a week day).