The Hadith is speaking of the spiritual heart. Humans consist of three faculties or components: body, mind, and soul. An ideal person is one who fosters all of the three. A person may be of the sound body while his mind and soul may be weak; while another may be strong in mind while the soul is weak.

The prophet tells us that the essence of man is his spiritual soul which outlives death.  It is not to deny the importance of body and mind. The last two faculties are not the ones that would bring him salvation in the next world; rather, it is thanks to the soul which is the locus of the divine spirit that God has dignified humans.

In conclusion, a person who neglects his soul becomes sick spiritually regardless of the state of his mind or body.

Allah Almighty knows best.