The video is the recommended route to take when racing Kai.
The earliest you can attempt this race is after you complete the Wawku Shrine.
First, before you even race Kai, you should hit up the forest. Bloom any throwing trees, etc. Get most of the obstacles out of the way so you don't get messed up when running.
Second, I recommend equipping Solar Flare. Helps with when/if you get frozen.
You may want to use brush skills. If you get frozen, slash can get you unstuck. When you reach the first three snowballs in the third part of the course, using the lantern to melt them really helps.
Don't bother with the time orbs; you don't need them.
Shortcuts are a MUST. I'm sure it's possible to beat her without them, but I couldn't and haven't seen a video of anyone who did. So using the shortcuts to your advantage is pretty much the key to this entire race.
You may want to race against her a few times to watch where she runs and to get an idea of where everything is. Getting this is basically skill and luck.