How to have the whistle warp to the ninth dungeon?

Short answer: it doesn't.

This appears to be another piece of misinformation in the manual. Another good example is the tip about defeating Pol's Voice (they hate loud noises, which implies using the whistle, but really was an allusion to the microphone peripheral that never got released in the States)

According to the wiki page for the "warp whistle" (also called the Recorder), its primary use is to uncover the entrance to the 7th dungeon, but will also take you to the entrances of completed dungeons. It will not take you to the entrance of the ninth dungeon.

Outside of dungeons, the Recorder has a number of other uses. It is the only means of revealing the hidden entrance to Level 7 in the first quest and Level 6 in the Second Quest, as well as several hidden caves in the Second Quest. When used in any other section of the Overworld, it summons a whirlwind which allows Link to warp around Hyrule, depositing him at the entrance to any other dungeon which he has successfully completed. The Recorder always warps Link to the entrances of First Quest dungeons, even during the Second Quest. For example, after clearing Level 3 in the Second Quest, the Recorder takes Link to where he found level three in the first quest. Level 5's location in the First Quest becomes a Fairy Pond in the Second Quest.