You need to install dkms and bcmwl-kernel-source.

Without internet connection

Both are located on the Ubuntu installation medium (possibly in different versions as reflected in the file names):

pool/main/d/dkms/dkms_2.2.0.3-1.1ubuntu5_all.deb
pool/restricted/b/bcmwl/bcmwl-kernel-source_6.30.223.141+bdcom-0ubuntu2_amd64.deb

Alternatively you can download the current versions:

1. Ask your friends to have them download the .deb files of the newest versions of the packages dkms and bcmwl-kernel-source. Both links lead to a list with the current versions of the respective package at the top for the right release (Trusty) and the right architecture (amd64).

2. Then copy these two files on your flash drive (1GB recommended).

Now, using the terminal, navigate to Desktop and type in this command:

sudo dpkg -i "drag one of the files here"

Then press enter. And click install if anything pops up. And you're done :)

With internet connection

If you can somehow establish a internet connection (e. g. through a wired ethernet connection), you just need to install those package normally:

sudo apt-get install dkms bcmwl-kernel-source

Installing linux kernel version 4.14 solved the problem (I wasn't aware that Ubuntu 16.04 used an earlier kernel).

Now keyboard and touchpad works.

I have not yet fully tested all special keys, but things like volume control and keyboard light works fine.