Large (6"+) trees on the dam, means large roots to decay....Muskrats may have only sped up what was probably inevitable in the dam leaking.

The least expensive, semi-permanent way to reseal the dam is to drain and rework the face of the dam with at least a 1' thick blanket of good clay highly compacted in 6" lifts...the thicker the better, and you should consider rip rap to frevent future muskrat damage due to the relative thinness of the dam seal.

A full re-coring/re-working to roemove all roots is the best repair....