If you must have an international brand hotel, within 20 minutes, your options are slim. The closest would be The Plaza or The Westin. And they're more like 30-35 minutes with an annoying transfer Line 9 to Line 1.

If you don't need an international hotel, try Lotte Hotel (official name Lotte City Hotel Mapo) in Gongdeok. It will be substantially cheaper in the 100-150 USD range. It's on Line 5, same as the Conrad. Both hotels are accessed from the subway stations.

The Gongdeok area has a good market and options in the neighborhood and is extremely convenient as the Airport Train stops here as well. Honestly, that area feels very local, too. It also connects to Line 6 (Itaewon). Line 5 also takes you towards Gwanghwamun and central Seoul for tourist sites.

If you have a little more money, you might try JW Marriott at Express Bus Terminal or the Ritz at Sinnonhyeon. These are both on Line 9 Express and take barely 10 minutes, but fair warning. Line 9 Express is continually 1,000%, push your way into the car crowded even at night. Mornings are no joke, sometimes having to wait for two trains or so due to the crowds. Local train is often way less crowded and only 20 minutes.

The other Hiltons and Hyatts are not really conveniently located within 20 minutes. Sure, taxis are an option, but with traffic in the mornings pre conference etc, time and money add up quickly. By subway, they require a few out of the way changes.

There is also Sheraton D Cube City at Yeongdeungpo, just two stops away (transfer at Singil Line 1). That being said, transferring to Line 1 can be sometimes confusing because the line splits off in many ways and being sure of which direction train could be a problem. It is just one stop, so it shouldn't be that bad really.