no, you do not want to use anything that has been precompiled to put into LFS, because you would end up with a myriad of random programs looking in all sorts of places for linked libraries; the reason you compile from source and/or follow the LFS book is to get your programs running nicely together
especially with the kernel.... you definitely just don't want to pull a non-standard kernel (such as out of mandrake) and slop it into a new distro... it wouldn't work unless you knew exactly how the modifications in the kernel will affect the rest of the system
if you just want to pull the kernel source from slackware's cd, you can do that (i don't think the kernel 2.4.22 would make any difference); but definitely compile it on your own
also, if you have a faster computer, why not just do some cross-compiling? ie, compile on your faster computer with the configuration of your slower pc... lots of guides on the net for that
and to finish: if you just want a small distro for a slow computer: just install one of your current distros (preferably slack) and just install only the things that are absolutely necessary; or, if you don't want to compile, download an older version of slackware so that it is more friendly with your slow pc (you can go as far back as slack 3.3)
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