I kinda feel like their relationship was defined by their first conversation, way back during the bridge plot when Katara was atoning for the whole Yeerk trial thing. If I recall correctly, Billy told her that a hero is someone who tries their best to help whenever they can.

They didn't become friends until months later, but sure enough: he had a hero on his side. It's rough sometimes, but he's incredibly grateful for the training. Even more for the companionship---and now, unconditional familial love. Billy's also grateful for her efforts in pushing him and Ronnie together, even if he complained along the way.

Her plight's still weighing on his mind.