Access password in iOS Keychain

Your Twitter app should not actually have your password stored on your iOS device. Rather it should have a security token, specifically an OAuth access token.

Have you tried resetting your password via SMS?

I don't know if the app in question stores the password in retrievable form in the keychain, but with iOS 7, you can enable keychain sync on iOS and have that information synced through iCloud to a Mac running Mavericks.

From there, you can inspect the iCloud keychain contents just like any other OS X keychain and obtain clear text password as well as any other data stored in the keychain entry.