Life In 19x19

Black 54 should be at G2. You need to understand the meaning of your thickness. White can not live by additional plays in the direction of 53. So you need not answer and have the opportunity to play from the other side. Put G2 on the board and look at the result.

jdl wrote:I did cap at K5, figuring that I couldn't kill the stone (it lived). My thought was to let white pick a side to escape from and profit on the other side while attacking his running group. In retrospect, I think I like J5 better since it's my weaker side...

This thinking is a major mistake. In the game situation it is absolutely necessary that you believe White can be killed. If White had invaded at K5 instead of K3, your thinking would make sense. Especially in a handicap game, it is unlikely that we could cap at K7 and kill White. However, in the game your walls are too deep. Go for the kill. Think hard about preventing White from making two eyes. If White lives, don't study how you might have done a better job on the outside, study what errors allowed such an unreasonable result. The day will soon come when your opponents will fear to play such unreasonable moves (well... except on Tygem anyway)!

Like ez4u said, your play after the invasion had a slack attitude. You seem to respect your opponent too much and assume he is a professional playing good moves. He is not. He is an 8k or whatever and most of his moves are rubbish*. If you can only realize which ones these are and not play as much rubbish as he does you will crush and destroy him. Here are the thoughts I would advocate (somewhat overcompensating for your timidity!):

After k3:
"Haha, you fool, too deep! You're gonna die!" or
"Haha, that move doesn't threaten either of my strong walls, I'm just gonna tenuki and play a bigger move, and then probably kill you anyway!"

After l2:
"What?! You expect me to answer that? I'm your opponent not your partner. Wake up and smell the coffee, Die Die Die!"

*Maybe rubbish is too strong for >50%, but certainly most are not optimal and at least a sizeable minority are rubbish.