TL;DR - Other than Bill's death, it seems that the only other death that truly affected the Bride was O-Ren Ishii's. That alongside many other clues lead me to believe that The Bride and O-Ren were very close, maybe even best friends during their years in The Deadly Viper Assassination Squad.
Re-watching Kill Bill Vol. 1, I noticed a lot of possible hints that point to a very close relationship (close to calling them best friends even) between O-Ren and The Bride.
Firstly, The Bride knows O-Ren's entire life story - the loss of her parents, how she killed Boss Matsumoto, how she became the top female assassin. Now one can assume that anyone in their type of business would know some of these facts about O-Ren's past, but she is the only assassin from both films whose background and current position is revealed extensively. The entire first movie is almost dedicated to her. The only other person on The Bride's list to receive as much development is Bill.
One of the biggest hints to their friendship was the scene after The Bride kills Gogo. "Silly rabbit, trix are for kids." This could be a simple play on words in reference to The Bride's real name, but it seemed that that little phrase held special meaning to them (e.g. an inside joke). Note, she did not share any witty banter with any of the other assassins.
Not only that, but the overall tone of the fight with O-Ren was a lot less vicious or aggressive. Granted they were fighting with samurai swords, but even their facial expressions weren't as angry, but rather somber. When they exchange words after The Bride lands her first blow, O-Ren pauses and goes on to apologize for ridiculing her earlier; not in the way Vernita Green "apologized" (begging for her life), but in a truly regretful and graceful manner. It should be noted that there was never any patronizing from The Bride, like there was with Vernita, or any aggressive "b*tchiness," like there was with Elle Driver. Both O-Ren and The Bride showed a mutual respect for one another.
Leading to the final hint: The Bride's reaction when she finally lands the killing blow to O-Ren's scalp. Not once did The Bride turn around to look at her triumphant victory. She is instead facing the opposite way panting as if she was trying to cope with what she had just done. There was even a moment where she seemingly starts to turn and face O-Ren's lifeless body, but she walks away instead. Why would a highly trained, cold-hearted assassin react in such a way?