Am I entitled to ask my wife to bear the burden of serving my mother, who will be visiting for ten days, or would that be a detraction from her rights?

If the wife agrees to host and serve her husband's mother, after the husband requests it from her, then this is a good deed and contains great good for the couple. For, honoring one's parents is among the best acts of drawing near to Allah, and the wife's assistance to her husband in honoring his mother is a righteous deed of great reward.