I don't believe there is an official president's badge at the Chapter or National level for Alpha Xi Delta.
Like other orgs. have mentioned here, a chapter can own a badge that is given to each president to wear during her term. My chapter had an antique one that was given to us during our installation.
The national president wears this:
It's hard to see but it's a gold medallion featuring our CoA with a president's gavel above it. Each link in the chain represents an established chapter of Alpha Xi Delta.
I saw it in person (at my Installation and DePaul's installation). It's pretty huge!