Learn Russian: Наш округ решил избрать её новым лидером. - Our district decided to elect her as the new leader.

The verb избрать (to elect) is a verb of appointing. In Russian, when you elect, appoint, or name someone to a role, that role takes the Instrumental case. So, the phrase 'as the new leader' is expressed simply by putting новый лидер into the Instrumental.

Since the English translation says 'as the new leader', shouldn't we use как?

No, you should not use как here. With verbs of appointing and naming like избрать (to elect) or назначить (to appoint), the meaning of 'as' is already built into the Instrumental case. Adding как would be a grammatical mistake.

Why is её in the Accusative case if this lesson is about the Instrumental?

её is the direct object of the action—she is the person receiving the action of being elected. The direct object stays in the Accusative case. It is only the new title or role (новым лидером) that takes the Instrumental case.

Why do we use the perfective verb избрать after решил instead of the imperfective избирать?

The district decided to achieve a specific result: successfully making her the leader. The perfective избрать focuses on completing this single, one-time action. You would only use the imperfective избирать if they decided to do this repeatedly or habitually.