For a key recovery attack, you'd basically need to break AES itself. There are no known practical key recovery attacks on AES (and if there were, it would not be considered safe to use), so your pretty much only hope would be to find some kind of side-channel attack on the AES implementation, or on the overall crypto framework it is part of.
Alternatively, if you suspect that the keyspace is small enough (e.g. if the key is derived from a passphrase which might not have enough entropy), you might be able to find it with a brute force attack. The chance of such an attack succeeding depends entirely on two things: the size of the keyspace you need to test, and the speed at which you can try the keys. The latter, in turn, depends on the computing hardware you have available (and, of course, your skill in utilizing it) and on the complexity of the key derivation process.
Padding oracle attacks, in any case, are specific to CBC mode, and do not allow key recovery. ECB mode, while insecure in other ways, does not suffer from them.