TimK's answer is right on point on how to kill white in the corner, but your question belies a more general concern: Thinking "I need to kill all the things" is a very common beginner's mistake.
Take a look at your final outcome; yeah, white lives. He lives small, blocked into the corner. You've got pretty much the entire lower left quadrant on lock, and you're poised to drive a wedge deep into white's right-side territory around G5 or diving into the top right corner and living there like he lived in yours.
Meanwhile, white is alive with three points.
As trade-offs go, this result really doesn't seem all that bad for black. If anything, I'd say black came out ahead here; territory-wise you're still about equal but black has a huge advantage in any future fighting since all of his stones are still ready to rumble while most of white's are locked up holding those three points in the corner. And black has sente, so how to bring this overwhelming force into play is entirely up to you. I really don't envy white here.
In other words, there's absolutely nothing wrong with letting white live in your corner, just make sure you're getting something at least as good if not better for the trouble.
(of course, if you can also kill it so you still have all your points and you also take all his points that's cool too)