Stain-grade wood repair for door mullion?
Unfortunately, I'll be surprised if you can effect a very successful repair.  Restoring the outer corners like those is tricky at best.  And having it stained only makes it worse, since the wood grain will be easily visible.  

Most likely any attempt to repair will make it worse looking not better.  I'd opt for something low-impact, such as a scratch-cover product like Olde English, which essentially re-stains the now-exposed wood fibers.  It at least conceals the bright colors of the underlying wood and makes the damage a little less noticeable.

But ultimately I'd confess to your landlord and let her decide how best to proceed.  It's her door, and her call.