I slaughtered a chicken, then found that a vein had not been cut; so I slaughtered it again!

Scholars differed concerning the ruling on an animal slaughtered without all its vessels severed:

Malikis: If the trachea (Ḳulqūm) or one of the jugular veins (wadajān) is left intact, it is not permissible unless the cut is quickly completed (equivalent to 300 arm-lengths if the wounds are fatal). However, if only the esophagus (marī’) is left intact, the slaughtered animal is permissible. Shafi‘is and most Hanbalis: The slaughtered animal is not permissible if the trachea or esophagus is left intact. It is permissible if one or both jugular veins are left intact. Abu Hanifa and a view among the Hanbalis (preferred by Ibn Taymiyyah): Severing three of the four vessels (trachea, esophagus, and both jugular veins) is sufficient for the slaughtered animal to be permissible.

Accordingly, leaving one vessel intact out of the four does not invalidate the slaughter according to Abu Hanifa and those who agreed with him, and the slaughter is considered sound and complete.