Holy shit how did you finish the level so quickly without the magic? I'm trying to figure out what I did differently. You even killed enemies with kunais more often than juggling them for bonus points. I don't get how you got so many natural stage points, especially since you lost your X4 multiplier near the second half of the stage. Good job nonetheless! I didn't even know you could pull chain the boss like that either! Is that just from charging your sword or something?
The main key is just not dying and not getting hit often. Its true I flubbed the second map pretty badly, but I got x4 at the beginning, lost it during that section, then got it back and kept it for the rest of the level, so I had it for large sections at a time. That x4 not only is giving me a bunch of extra points but it is keeping my run speed at 130% normal helping me finish the stage faster.
In addition to losing the multiplier, getting hit and dying takes away a TON of points and tacks on extra time to recover from the mistake. The difference between this run and previous ones where I did not get S is that I previously died at least once from falling in a pit and ended up taking much longer and having a much worse score such that even when I hit the time bonus I didn't have enough points to make it.
So I guess the lesson here is to concentrate more on not getting hit and on moving quickly than on doing juggle combos. Use the combos to get to x4 ASAP but after that only bother when you have an easy target and not much threat of getting hit while bouncing them around. Otherwise kill enemies as efficiently as possible and move on.
Regarding the boss fight...
If you fully charge your sword attack it pierces enemy defenses, including Yuki Onna's ice shield. You'll notice I also used it on the guy that swings the big polearm around to kill him instantly (the other method is to wait until his attack finishes for him to be vulnerable). You can do this either by holding down the button for long enough or pressing forward+sword immediately after a parry.
So I started the Yuki fight by doing a fully-charged slash, followed it with the contextual attack during her recoil (tap Chain button) and turned and hit her with a couple kunai. Then I parried her ice shards and used the counter-slash to again go through her shield and repeat. Combined with the double-damage power-up she dies extremely quickly with this method. I'm pretty sure its the fastest possible way to kill her.
Edit: It probably doesn't matter much but it may be worth knowing that you get bonus points if you collect something that you can't use - chicken when health is full, scroll when already have magic ready, attack power-up when already have the boost active, etc. That's why I bothered to get things like that when they weren't too far out of the way. Kinda wasted my time getting the extra lives though, didn't help me any...