She won by virtue of her voice and performance capabilties.
The music she used was used WITH PERMISSION from the artists who wrote them. From what little I have seen, this music is USUALLY about 10 or more years old. The artists have re-couped any investment they may have made in the music. "Heard it on the Grapevine" is one such example.
On the other hand, Kelly is just STARTING her career; with music she created. For Simon to claim that Kelly is somehow wrong to withhold her work until SHE can profit from it is disengenuous at best, blackmail at worst.
Why does she owe Simon (who has a personal vested interest in the profit of this show) the work she has done outside of the show? In what other field would this even be a question??
If she were a playwright, would her plays be open for Simon to use? If she were a plumber, would he demand she plumb his house for free? If she were an actor; would she have to perform for any show he wanted for free?
Winning a contest based on talent does not mean you are now subject to indentured servitude to the contest sponsor.
She writes a play that is a box-office success. The theatre where this happened now has the rights to stage productions of any future plays she happens to write. That seems fair.