Learn Italian: La televisione non funziona. La controlli, c'è un guasto. - The television does not work. Check it, there is a fault.

You are thinking of informal commands (like telling a friend controllala, meaning "check it"). However, this sentence uses the formal command, as if addressing someone as Lei. With formal commands, object pronouns like La must always go before the verb.

Why does controlli end in -i here? Isn't that the tu form?

In the regular present tense, yes, tu controlli means "you check". But this is a formal command. For verbs ending in -are like controllare, the formal command form ends in -i. If you were giving this command informally to a friend, it would actually be controlla.

Why use funziona instead of lavora for "does not work"?

In Italian, lavorare is only used when a person or animal is actively doing a job. When talking about a machine, device, or system functioning properly, you must use the verb funzionare.

Are the two words La in this sentence doing the same grammatical job?

No, they have entirely different roles! The first one in La televisione is a definite article meaning "the". The second one in La controlli is a direct object pronoun meaning "it", referring back to the feminine noun televisione.