As the "extension" doesn't form part of the original lease, and hasn't been incorporated into the lease, and you haven't been asked to sign a new lease incorporating the unauthorised extension, he has no right to ask you to do the repairs in order to fix the leaks into it, but as it's the existing structure that's leaking, and not the unauthorised extension, then you are obliged to maintain the structure of the property, so you're stuck I'm afraid. The question becomes, can you defer the repairs, as you are being asked to do them for an immoral reason.
The question now become, exactly where the leaks are, and what's the cheapest way if fixing them. How does a fibreglass pan on your balcony sound.