I don't think you can fill and sand that big of a scratch. It will for sure be noticeable. Even the tiny little scratches cross over the natural grain of the wood and stick out like a sore thumb. The fill job you could do here would just look like a bunch of scratches filled and stained.
The best plan is if, like speede said, you can get a piece from a closet to replace the damaged piece and then replace that piece with a close filler. You'll probably need a hardwood person to do that sort of repair. Actually, you might want to get a hardwood person to look at it anyway because they often know where to find the exact wood you might need. I'm not sure about Ontario but in Alberta this is beyond wear and tear and the landlord would be reasonably allowed to repair it which could mean sanding and refinishing the whole floor.
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