Thank you for The tips AndyBall_uk!
I did try another Apple branded keyboard, without success. However, I did manage to creative a new admin user through my guest user account, exploiting the initial admin user account details. Although, my time machine told me my HD hadn't got enough space, so in the end I got hold of a reinstall programme from a friend who also has a Mac, using that to start from scratch again. Doing it right now, and it seems to be working!
It got me on the trail of something, which I am grateful for!