These old exercises you mention are Hanon exercises. There is a very comprehensive book for technical exercises, called 'Hanon - The Virtuoso Pianist'. The very first five finger exercise in that book is the one you mention, however, there are many, many more. The first one is the most universally known, and the one that most people think of when you say 'Hanon'. It certainly is useful, but the exercises in the book progressively increase in difficulty, and are designed to increase the player's technique as they progress through the book.
In regards to your daughter, I must apologize, but being from the States, I'm not sure what you mean by 'A Level'. In searching, the best I could come up with is that it appears to be a fairly advanced, high school level music program, but I don't know any other details besides that. Please correct me if I'm wrong about that.
However, if she is fairly advanced, purchasing the entire 'Virtuoso Pianist' book will serve her well if she is serious about playing piano proficiently. And yes, for the same reasoning as was given for Jordan's 'Five Finger Pattern', it would serve her well to progress up chromatically through all the keys, using the same five finger pattern. Doing the minor keys for the Hanon is probably a bit of overkill, though.
I should also mention that the exercise continues down. So, after you complete the ascending octave with B, D, E, F, G, F, E, D, you continue, without rhythmic break, descending with G, E, D, C, B, C, D, E, F, D, etc. And, it may be obvious, but I have to mention that all of these exercises are to be played with both hands. However, it never hurts to start playing any exercises like this with separate hands, to insure that every finger is playing strongly, and that the whole exercise is executed evenly, with equal volume and tone on every note. When you're sure that each hand can do this separately, then both hands can be put together - but always listen for that evenness in both hands.
It may also be wise to play the first Hanon exercise in 'C' only, first, until it can be played very fluidly, in a relaxed, confident manner, before transposing up chromatically. It is much more challenging to do chromatically than the five finger exercises that Jordan demonstrates in the video.
I really do recommend that video, though. He demonstrates a lot more exercises, and gives good advice on how to properly execute each one.