I don't fly too much so I don't have too many miles. Since I won't be able to get an award ticket, I'm trying to get the most out of my money. I was able to get a free stopover in Tokyo for 2 days with JAL on my way to Seoul. I was wondering if there is something similar for the LAX to TPE route? I've tried Air China since they have layovers in Beijing but they charge an additional $400 if I want to turn the layover into a 2-3 day stopover. I would like to see Beijing but am open to other cities. My schedule and my small collection of miles are listed below:

4/18/2014 LAX to TPE
4/30 or 5/1 TPE to ?
5/3 or 5/4 ? to LAX

Ultimate Rewards ~50000
AA 11391

Any help would be appreciated.