Lyrics translation must account for rhythm, rhyme, and singability in the translated lyrics. In this study, we focus on singability and investigate effective prompting methods for translating singable lyrics, including verification-guided and multi-round prompting, applied to large language models. First, we curate a multilingual lyrics translation dataset covering a total of six language directions across Chinese, Japanese, and English. Next, we evaluate seven prompting strategies, with instruction complexity increasing incrementally. The results show that multi-prompt strategies improve singability-related aspects, such as rhythmic alignment and phonological naturalness, compared to naive translation. Furthermore, human evaluations using songs created from translated lyrics suggest that moderately complex prompting strategies improve singable naturalness, while more complex strategies contribute to greater stability in perceived quality.