why doesnt Sesshomaru use the tensaiga to Reattach his left arm

well since sesshomaru's arm got cut off kwa tetsaiga, he cant use the tensaiga to fix his arm. only because both swords are rival swords and both of the swords came from the same fang.i hope this helped:)

because sesshomaru's sword can only save a hundred lifes

Well my theory about why he never tried to use the Tenseiga to restore his arm is: one he respected being defeated and accepted the loss, also after accepting that the use of arms from other demons were of no use to him, and, two because his left arm got left behind in the grave, and probably never imagined it would re-manifest as seen in the episode when the Bakusaiga is born when his arm returned with the blade he seemed pretty surprised.

Now as far as the sword being able to heal his arm, personally I don't believe the sword works that way. The Tenseiga is a sword of Heaven, nick named the healing blade, it's power taps into the other world it's not a material blade so I do not believe it would just heal the flesh if it didn't involve restoration of the soul and au bringing someone back to life.
Could be the reason why it was unable to help Kagura in her final moments, because she wasn’t dead yet.

Perhaps the Tenseiga is only effective on the deceased so long as their body remains, dead souls and spirits of the other side.