Was it actually necessary to compute |w|?
No. It cancelled out of the final result.
You don't have to compute the length of w. It cancels out before the final answer. And, all you need is the square of the length of v, |v|2. Some books show formulae for projection that make use of these facts (but, to my taste, are less intuitive).
Here is another example, this time not so easy. The vector w is represented by by (3.2, 7)T. The vector v is represented by by (8, 4)T. Find kv and u.
Does w = kv + u, as it should?