Summary: The gravitational and electromagnetic (EM) fields of the Dirac electron are described by the Kerr-Newman (KN) solution. We elaborate a regular source of the KN solution which satisfies the requirement of flat space-time inside the source and realization of the exact KN solution outside the source. This requirement removes the conflict between gravity and quantum theory and determines many details of the source structure. In particular, we obtain that the KN source should forms a gravitating bag model, similar to the known MIT and SLAC bag models. As opposed to the known bag models, the self-interacting Higgs field should be confined inside the bag, while outside the bag the gauge symmetry should be unbroken to provide the external KN gravity. We show that the twistorial structure of the Kerr geometry (the Kerr theorem) determines the Dirac equation structure, resulting in a variable mass term which is generated inside the bag through interaction with the confined Higgs field. Similarly to the other bag models, ellipsoidal deformations of the KN bag creates a string-like structure of the dressed electron – a circular string located along the perimeter of the KN bag.