Building an RLC circuit will not work, at least not very well. You will need a 2.4 GHz antenna that is fed into your circuity. This is a lot of work for someone inexperienced, and can even be a lot of work for someone who IS experienced. You will also have issue determining if the signals you are seeing are coming from the keyboard, or a wifi router, or from something else around. Overall, I would not recommend going down this route.
As far as I can tell, this keyboard is not using Bluetooth. If they were using bluetooth, I am sure they would advertise it as such since this is a big selling point for some keyboard (due to interoperability with built in bluetooth). So I wouldn't worry about going down the path of sniffing bluetooth.
Clabacchio linked to some packet sniffers, a quick look at them looks like they can sniff 802.15.4. 802.15.4 is a standard that is used for many things at the physical layer and media access control layer. There is a decent chance the Microsoft would have built their technology on top of 802.15.4, but no promises. If they did use it, then just pick yourself up one of those sniffers and start capturing data. It will take a lot of trial and error of typing a key and seeing what is captured, and just keep doing it and see if there are any patterns. Where problems will arise is if they are encrypting the signal before transmitting it, it would be very difficult to determine anything at all. People devote their lives to trying to crack encryption. It usually take years of work before anyone cracks commercially used encryption. If they are encrypting the signals, you might get lucky and figure out they are using weak encryption, but if not then you should just give up on trying to sniff it.
The other problem you might run into is if they aren't using a standard protocol, you would need to interpret what the signals being sent actually mean. This is a long process that I would not recommend. But if you wanted to go down this path, you would best off to get yourself into an environment with no other 2.4GHz signals around so that you know that every thing you see at the frequency is relevant to the keyboards communication. You would have to follow a similar process as I mentioned before where you press a key and see what you receive, but instead you would be looking at most likely phase changes in a carrier signal and interpreting what those mean as far as 1's and 0's and then would have to interpret what they meant to the keyboard.