That iPod client would be a better object for hacking then, I think :-) (no, I don't think this is feasible for even a bunch of developers to do that in their spare time). It is fit for the right platform...
The Linux client page doesn't really mentioned it, but last time I looked (warning: older version, some time passed since then) they had only x86 (i686) clients, not even x86_64/amd64 - let alone ARM (ARM6!) which we would have on the Kindle. Not to mention that there isn't enough CPU power (not close to 1 GHz), not enough RAM (well, we would have exactly 256 MB, but it's even more expensive since we do not have Swap).
I don't think this idea is feasible. Probably "easiest" is to port Android (which has Skype clients, I think?). And that's far off from what I would call "easy" at all. Otherwise it's a proprietary protocol, encumbered proprietary software - not what I would call hobbyist developers' beloved plaything. The only chances to see something like Skype on the Kindle I can see is to persuade Skype and Amazon that this would be something they could make money and happy customers.