The version I'm presenting here (for now) is actually the double fire route for 5-1 and 5-2. I strongly recommend double fire for 5-1 and 5-2, because single fire is only a little bit faster while being significantly more difficult.
Combine for double fire using the two Bobos and follow the route to the end of the room.
In the second room, don't use double fire up the slope. Because of the long recovery animation, it turns out to be faster if you just run to the ladders. When climbing up the ladders, jump off the top of each of them onto the next one by either grabbing the ladder's ledge or doing a small jump when you're near the top of the ladder. When triggering the Waddle Dee cutscene, also don't use double fire here. Like the minecart autoscroller, jump repeatedly to save time. Shard 1 is on top of an igloo after a few food items.
The Chilly miniboss becomes invulnerable when attacking, and even with double fire, you won't make it there in time before it attacks. In the meantime, break the ice stalactite with double fire before doing two double fires to kill the miniboss. If done correctly (i.e. three hits per double fire), the second double fire will end before the door opens, allowing you to float up to collect Shard 2. If that's too risky, instead of doing a double fire to the right, you could also do three double fires to the left (reminiscent of the single-fire strat). Since the miniboss's attack doesn't last that long, one small timesaver is to use the tail end of the double fire to break the ice so you don't have to float as high up or fall as far down.
In the following room, fire across to the second hole in the ice (past the third pillar) and drop in. The pillars are all evenly spaced, so the same double fire setup will work. In the water, swim to the right and double fire through the hole. Collect Shard 3 in the alcove in the bottom right and go back the way you came. To line up the double fire to the left, use the height of the rightmost enemy.
Double fire to the end with the route, but note that it's pretty precise. There are alternative routes that involve floating up close to the top of the screen, but these are significantly slower. The jump between the floating platforms in the second clip can be done more safely with a puff, and jumping to the next platform makes it easier to clear the Putt's boulder.