They've burned through several contractors, and now you're the latest one to be headed for the auto-da-fé?

Unless you're can't-pay-your-bills broke and there are no other jobs in your area, the correct response is "I'm sorry. This can't be fixed. It needs to be scrapped completely and replaced with something maintainable. If that's not an option, I'm going to have to resign."

Even if you manage to fix it, it'll still be a pile of crap, and you'll never get the credit you deserve.

On the other hand, if you don't manage to fix it you're going to get blamed.

Zero credit on the upside, major blame on the downside. There's nothing for you in this but pain, my friend.