When you have the needed packages on the slow machine in the /var/cache/apt/archive directory you need to do 2 things.
One disconnect the slow machine from the internet.
Two, instead of running "apt-get install <package name> run;
apt-get install --reinstall <package name>
This will insure that it will look in the right place for the package.
This will not work unless all dependencies for the package to be installed are also in the /var/cache/apt/archive directory.
I thnk a better approach would be to check out the package "aptoncd" on your fast box. You can study that and I have not ever used it but you should be able to create a "repository" on a disk and use that to install the needed packages on the slow box.
This is probably done by adding the cdrom line back into your active /etc/apt/sources.list but I am not at all sure of that so read the man page of aptoncd.