What is the correct way to spell refuah shelimah in Hebrew. Is it with a שלמה or שלימה?
Different conventions for spelling are used for different purposes.
Siddurim (prayer books) use ktiv menuqad. Vowels are mostly specified by dots, not letters. The form שלמה would be used in a siddur, and the Tzeirei vowel would be indicated by dots.
Modern Israeli newspapers and books use ktiv male. In this system, a lot of Vav's and Yud's are inserted, to help indicate to the reader the vowels to use. In a newspaper, they'd spell your word שלימה.