I'm a big fan of Sachi mostly because Haqua.. err, Saori Hayami voiced her. A part of me wanted her to whip out a scythe for a weapon!
I imagine the reason why she didn't sing Red-nosed Reindeer was for one of two reasons: Either she couldn't sing in English very good, or it would've cost the studio more money to hire her for the single episode role she has. Humming would be much easier(read: cheaper) for a seiyuu than singing.
Sachi also used an invisibility cloak in the LN when Kirito came upon her in the sewer, which apparently made her impossible to find without Kirito's tracking skill. Too bad she didn't whip it out in the trap room!
