Ich bin kaputt is rather colloquial, it's literally "I'm broken". A more formal translation would be Ich bin erschöpft.
I'd likely understand that if I heard it. I would think it would accept "Ich bin müde". I know müde is tired, but if you're exhausted, you're tired. Hmm...
I said, "Ich bin todmüde," which means "I am dead tired" (the same thing as "exhausted" to an English speaker). That should have been accepted.
"Everything below the waist is kaputt!" - Lili von Schtupp
Can't we say "Ich bin leer" like in french "Je suis vidé"?