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There is a commandment to obliterate Amalek.
Shaul Hamelech was commanded to obliterate Amalek and failed.
Dovid continued in the process, but obviously didn't destroy them completely.
None of the later kings of Israel (including Chizkiyahu who was a righteous king, Shlomo who was a world power etc.) even went to war against them.
Why was this commandment never fulfilled?
Per the Pesikta Rabosi Parsha 13 - Once Shaul was the king, Amalek can only be eradicated completely by the children of Rachel. The other kings were from the children of Leah.
Insofar as it was necessary for Amalek to be destroyed before the Temple was built (Sanhedrin 20a, Rashi ibid) it seems that this mitzvah was fulfilled. Shaul did not fail, 1 Samuel 15 is clear that he destroyed the people as was commanded but he disobeyed by sparing King Agag [only] who was subsequently executed by Shmuel haNavi. With regard to later mentions of Amalekites it is perhaps along the lines as mentioned in the comments that there were remnants which no longer constituted "the nation of Amalek" or alternatively they were members of related groups not included in the mitzvah:
Guide for the Perplexed 3:50 The list of the families of Seir and their genealogy is given in the Law (Gen. 36:20-30) because of one particular commandment. For God distinctly commanded the Israelites concerning Amalek to blot out his name (Dt. 25:17-19). Amalek was the son of Eliphas and Timna, the sister of Lotan (Gen. 36:12). The other sons of Esau were not included in this commandment. But Esau was by marriage connected with the Seirites, as is distinctly stated in Scripture; and Seirites were therefore his children; he reigned over them; his seed was mixed with the seed of Seir, and ultimately all the countries and families of Seir were called after the sons of Esau who were the predominant family, and they assumed more particularly the name of Amalekites, because these were the strongest in that family. If the genealogy of these families of Seir had not been described in full they would all have been killed, contrary to the plain words of the commandment. For this reason the Seirite families are fully described, as if to say, the people that live in Seir and in the kingdom of Amalek are not all Amalekites; they are descendants of some other man, and are called Amalekites because the mother of Amalek was of their tribe. The justice of God thus prevented the destruction of an [innocent] people that lived in the midst of another people [doomed to extirpation]; for the decree was only pronounced against the seed of Amalek. (Freidlander page 380-381)
