Ice Ball Skip
Normally, after the final outdoor area in Wawku Shrine (the cannon room), one must release an ice ball, then use Galestorm to blow it into the room filled with burning hot, fast-spinning gears. Blizzard is then used from the ice ball to the gears in order to freeze them so Ammy can platform around the room to obtain the key and break the lockjaw.
Interacting with the ice ball can be entirely skipped by either using Snowstorm (aka Snowflake) or by simply platforming around the gears.
Start by getting close to the hot gear directly in front of the entrance to the room. Use inner camera to get the brush camera closer to the gear. Brush and draw a moderately long stroke down, a long stroke right which crosses the top half of the first stroke, and finally a stroke going diagonally down right and through the intersection between the first two strokes. This should be recognized as Snowstorm, which will freeze the first gear. From here, simply hold the brush button and wait for the cooldown to expire.
Once the brush comes up, you can use the frozen gear as an ice source to Blizzard the other gears that make up the path to the right platform where the key is located. CB the key and pick it up. Quickly turn around and use Blizzard again over all the gears that make the path to the large platform where the lockjaw is. If you are too slow, the gears will shake, then thaw, forcing you to use Snowstorm again. Jump to the lockjaw and proceed.
Note that the recognition on Snowstorm is probably the least consistent in the game, so it will take some practice to make it work reliably.
This method is a couple seconds faster, as it avoids waiting for Snowstorm cooldown, but is easier to lose time on. You must maintain gold dash all the way through to make these jumps.
First turn left after entering the room. Jump to the platform with the burning chest (Emerald Tassels). Being careful not to dash into the platform walls or to turn too sharply, jump off the right corner of the platform behind the chest. Double jump and air tackle for distance, aiming for the edge of the platform with the lockjaw. If you accidentally run into the chest, jump to restore your dash, circle around to build momentum, and try again.
From the lockjaw platform, make a loop by turning left 270 degrees to build momentum. Jump straight off the platform going toward the platform with the cracked floor. Double jump and air tackle. As you land, CB the floor. Ammy should fall into the pit. Grab the key, then jump to the right. As Ammy lands and restores gold dash, make a tight circle and jump back to the lockjaw platform.