I still haven't gotten this working.
I don't know what libz version kopete requires. apt-get says libz or zlib has been included in zlib1g, which I have installed. I can also see that I have the libz.a, libz.so, libz.so.1.1.4 in my /usr/lib/. Also, in my /usr/include/ I can see zlib.h.
As far as dev packages go, I have zlib1g-dev, which claims to include libz-dev. I recently installed libc6-dev, kdelibs-dev, kde-devel, kde-designer, kdebase-dev, and all associated packages. None of these made a difference.
So I downloaded a new kernel (2.4.22) and installed it. I noticed a libz/zlib section which was turned off by default, which may have meant it was turned off for my previous install. This definitely got my hopes up. But it still made no difference.
I've noticed quite a few postings on various forums about similar problems with libz. Nothing that seems to do it though.
Thanks for the help so far. Very much appreciated. Let me know if you have any other thoughts.