There is a common, though possibly dated idiom 'like a streak of lightning' which shares precisely the meaning of 'like a streak', and exists in subtle variations 'like a blue streak of lightning', like a greased streak of lightning' or 'like a streak of greased lightning'. Indeed the Oxford Dictionary of Idioms gives the idiom as:
like a streak (of lightning) very fast. informal
By way of further backing up that Wodehouse himself used the phrase as a shortened version of the full idiom, we can see that he used used the term 'like a streak of lightning' in much the same way in his earlier work 'The Man Who Disliked Cats' which predates Thank You Jeeves by some 22 years.
Nor did he move, till I ’ad seized the parrot and replaced him in the cage, when he shot upstairs like a streak of lightning. By sheer force of character that excellent bird ’ad won the bloodless victory. I drink to ’im!”
and later in Ring for Jeeves, published almost 20 years after Thank You Jeeves:
I’ve watched the animal run with my own eyes, and it’s like a streak of lightning. All you see is a sort of brown blur.
Therefore the full understanding of the term is that he moved not only fast, but as fast as lightning.