The space is doing the work here. Closed up, it describes something — everyday clothes, everyday problems. Split apart, it tells you how often. Advertising has blurred this badly: "everyday low prices" is technically correct as an adjective, while "we are open everyday" is not, and seeing the second often enough makes it look normal.
An adjective meaning ordinary, commonplace, or routine. It sits before a noun.
Two words. An adverbial phrase meaning each day, daily.
Quick way to remember
Try inserting "single" between the two parts. "She checks it every single day" works, so it is two words. "Every single day clothes" is nonsense, so that one is the adjective and closes up. If the word sits in front of a noun and means ordinary, it is one word.