I could buy a bag of cement and get a tiny bit out and
mix it with a bit of sand and so on, but that would be
a pain in the neck for such a small job, and it is hard
to mix accurately such small quantities. And I would
have to chop off the old mortar to make space for
the new.

You can buy bags of ready mixed bricklaying mortar (i.e. just add water).
Still 25kg bags, but at least you don't have to measure it out, and they're
pretty cheap. If you're lucky you might find a 10kg (or even 5kg) bag, but
I'd be surprised.

Christian.