I think we should use Native Korean numbers in that case 13 is actually yeolseh 엘셋.

자는/ 나는 13 살 이에요. chanun/nanun ship sal ee-eh-yo.

나는 열세[13] 살이에요 or 저는 열세 살이에요. Here's an old song titled "나는 열일곱 살이에요" (I am 17 years old) in case it might help. https://www.youtube.com/watch?v=77Y3LtpiJhI It goes "나는 가슴이 두근거려요 / 가르쳐 줄까요 / 열일곱 살이에요 ...". My heart is pounding / Shall I tell you / I am seventeen years old ...