The answer is likely to be yes. For starters, kayaking on the Han river is a commonly practised water-sport both by local clubs, and by tourists with organised tours. Park authorities even rent kayaks out. Now, the water quality of the Han river has been radically improving since governmental efforts began making it a priority in 1982. The most recent published analyses I could fin date from 2011, and present positive conclusions stating that, at the time when the article was written, the river is clean to be used in households and swimming pools if purified:
The Biological Oxygen Demand (BOD) level measured at 10 spots along the river was 2.3 milligrams per liter on average, they said. The previous record was 2.33 milligrams per liter established in 2003.
BOD is a chemical procedure for determining the amount of dissolved oxygen needed by aerobic biological organisms in a body of water to break down organic material present in a given water sample at a certain temperature over a specific time period. It is not a precise quantitative test, although it is widely used as an indication of the organic quality of water.
Last year's BOD level shows the river's water is clean enough to be used at homes and swimming pools with normal water purification procedures.
In addition, this 2014 report on Seoul sewage from Seoul Solutions even states that the Han river is clean enough for swimming:
Swimming in Han River made possible by improved water quality
Seoul has currently achieved 100 percent connection of its population to the sewerage system. The total treatment capacity, originally 5.81 million ㎥/day, was reduced to 4.98 million ㎥/day by the end of 2013 due to the conversion into advanced treatment facilities with higher nitrogen and phosphorus treatment efficiency. The total sewer length is maintaining 10,487km in 2012. With the introduction of advanced sewage treatment for nitrogen and phosphorus removal since 1996, the water quality in Han River was further improved; BOD in midstream and downstream was reduced to as low as 3mg/L or less, which makes swimming possible.
Up to you to decide on whether or not the water is clean enough for kayaking. Regardless of what online report say, personally, I would avoid drinking it.