We analyze an equilibrium problem for 2D elastic body with a T-shape thin inclusion in presence of damage. A part of the inclusion is elastic, and the other part is a rigid one. A delamination of the inclusion from the elastic body is assumed, thus forming a crack between the elastic body and the inclusion. Nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces. The damage is characterized by a positive parameter. The paper provides an asymptotic analysis of the solutions as the damage parameter tends to infinity and to zero. A passage to infinity of a rigidity parameter of the elastic part of the inclusion is also analyzed. Junction conditions are determined at the connection point between the elastic and rigid parts of the inclusion. An existence theorem is proved for an inverse problem of finding displacement fields and the damage and rigidity parameters provided that a displacement of the tip point of the inclusion is known.