I think the most hassle-free option will probably be Asiana from LAX (with your IAH-LAX flights on UA/CO metal). They have both daytime and red-eye flights on the LAX-ICN route, plus the option to connect through ORD or other gateways, and that would give you a lot of scheduling flexibility, especially if you are good at sleeping on planes.

Another option would be to take UA IAH-NRT-ICN and vice versa, which would give you an international seat all the way. They only offer daytime flights on this route, so you would lose two days going to Korea and one day coming back, and of course their service is not nearly as good as either of the Korean airlines. If you have to fly coach, this is probably the better option since you can get into Economy Plus (more legroom) as a frequent flyer.

Either way UA is probably a good program for you, since you live in a UA hub and could take advantage of E+ and upgrades on any domestic personal flights you need to take, and would also have many convenient redemption options for international flights.

Delta and/or Korean Air would be the other realistic option, but the Delta and Korean Air programs are not nearly as generous as United's, and you would also have to go through DTW or ATL to get anywhere.