I hope you've been able to solve this already, but here's a couple of things to think about. First, in ePOS-Print_DK_iOS... manual that I'm looking at, the TM-T20 is not listed as a valid printer type for initWithPrinterModel. The valid values it lists are: TM-T88V, TM-T70, TM-P60, and TM-U220. Maybe try the T70? Or maybe it's not compatible with the ios sdk?
Another thing to try is to see if you can access the web-based config utility for the printer. Go to
http://192.168.0.10
(I'm assuming yours has one, I'm using a TM-U220B.) If you can't reach the config web utility, you probably can't reach the printer.
There's probably a way to dump the ip config of the printer. (On mine, I hold in the "stupid-little-button" on the ethernet interface for 3 seconds.
It's also possible that it won't work with the USB -- the airport might not include that in the IP network. I don't know too much about that.