In his "Treasury for Word Lovers," Morton Freeman laid down a workable rule 20 years ago: Spell it as one word after a verb, two words after a preposition. By that rubric, reader Kline had it right -- he was staying awhile in Chicago, and also staying for a while. The rule works for me, but you should be aware that Merriam-Webster characteristically opts for anarchy: "Follow your own feel for the expression, and write it as one word when that seems right and as two words when that seems right." The distinction "is not important at all."
Before dropping the topic, let me rant once more about the abuse of "while" in the sense of "although" or "because." Consider two Horrid Examples from The New York Times a year ago: "Mr. Handwerker said that while (read 'although') he agreed with the reasoning behind the government's timetable, the company's engineers would find a way ..." And in the same article: "The contractors say that while (read 'because') they can move more quickly, they are pleased to operate ..."
What about "all right" and "alright"? For a good many centuries the two-word version was the only version. In the early 1900s, "alright" crept into newspapers and popular magazines, but it was not until 1934 that it passed through the heavenly portals of professional lexicography. Why the wait? The language had long welcomed such respectable unions, e.g.: altogether, albeit, already, although. Theodore Dreiser put "alright" to work in his manuscript for "The Genius" (1915). James Joyce used it in Molly Bloom's interminable soliloquy (1922). It strikes me as an inoffensive melding -- even though The Associated Press Stylebook emphatically disagrees. The choice lies in a writer's ear and eye.