I did receive a response from the Mid-Ohio Time Warner Cable customer service office. Apparently, many people contacted their offices to complain about missing the last episodes of Dae Jo Young. Time Warner will broadcast the last 4 episodes of DJY on channel 24 on March 24th, 25th, 31st, and April 1st starting at 9pm on each of those nights.
At this time, Time Warner Cable has no plans to add any new Asian TV channels (with English subtitles). However, since they seemed to hear viewers about DJY, I'd suggest contacting Time Warner (service@twcmidohio.com or call (614) 481-5050) and let them know you'd like KBS America (w/Eng. subs) or some other similar channels.