I am not aware of any public crypto library with countermeasures against side-channel attacks based on power analysis against AES (or any other symmetric cipher). There are 2 good reasons not to even think to write them:
- Loads of techniques are patented. You may receive a take-down notice very quickly.
- With power analysis, you must have physical access to the device and in most cases you also need to be able to run arbitrary code on the device. For smartphone apps, it is much easier for the attacker to lift the key via traditional means (e.g. jailbraking).
Simply put, don't use important AES keys in an Android phone (or iPhone) if your attackers have physical access to it. Note that in the video the first attack was mounted on a weak implementation of ECC, not on AES.
You may prefer to address more likely side channel attacks where the attacker operates from remote, like timing attacks, cache attacks, or padding oracle attacks.
Effective countermeasures can be done at the protocol level, independently of the AES library. Otherwise, you could consider using the Android NDK and the NaCl library.