The Kac-Moody central extension can be described in terms of algebraic $K_2$. This was first discovered I think by Spencer Bloch in the early '80s. There is a scattered literature that spells this out in different contexts - the main published references I can think of are by Deligne-Brylinski (Central extensions of groups by $K_2$) and the papers it cites by Deligne (in particular Le Symbole Modéré), the papers by Brylinski-Mclaughlin on the Segal-Witten reciprocity law and symbols etc. (I learned of this from the famous unpublished manuscript of Beilinson-Kazhdan, I think it appears also in later published works of these two individually). Anyway this gives a formula for the Kac-Moody central extension in terms of the tame symbol. Actually one place where the whole story is spelled out beautifully is Kapranov's paper on Eisenstein series and S-duality.
To summarize briefly: $H^4(BG,Z)$ actually consists of algebraic cycle classes, i.e. it's equal to $Chow^2 (BG)$. Bloch showed (in the 70s) that this is the same as $H^2(BG,K_2)$ and used this to give a beautiful picture for second Chern classes. Anyway this can be interpreted as central extensions of $G$ by $K_2$. Now if you're over a local field (Laurent series say) the tame symbol is a kind of residue map, taking $K_2$ of the local field to $K_1$ (i.e. units) in the residue field. So you can push out the $K_2$ extension to get a $C^*$ extension of the loop group, as desired. While $K_2$ is an intimidating beast, this gives an explicit formula I think since the tame symbol is explicit... but I'm the wrong person to give you that formula.
BTW for the multiplicative group this ends up giving a POV on Weil reciprocity, that was spelled out by Witten in his gorgeous paper on Grassmannians, QFT and Algebraic Curves, and is explicated in a paper by Brylinski with a related title (Central Extensions and Reciprocity Laws) and most recently in a very pretty paper of Takhtajan.
