Learn Italian: Non c'è nessuno specchio nella stanza.

Because the masculine noun that follows it, specchio, starts with an 's' followed by a consonant ('p'). Just like you must use the definite article lo before specchio (lo specchio), the adjective nessuno keeps its full 'o' ending before masculine words starting with 's + consonant', 'z', 'ps', 'gn', or 'y'. Before most other masculine words, it drops the 'o' and becomes nessun.

Why is there a non at the beginning of the sentence if nessuno already means 'no'?

Italian requires a double negative when the negative word (like nessuno, niente, or mai) comes after the verb. Since nessuno comes after the verb c'è (is there), we must put non before the verb. To an English speaker it feels like saying 'There is not no mirror', but in Italian this is perfectly correct.

Can I use the plural and say nessuni specchi to mean 'no mirrors'?

No, nessuno and its variations are almost exclusively used in the singular form when acting as an adjective for countable things. Even if we might say 'no mirrors' in plural in English, Italian keeps both the adjective and the noun singular: nessuno specchio.

Why is it nella stanza instead of just in la stanza?