solar system - How long should it take for us to observationally determine if Caltech's Planet Nine is really there?

Those are model calculations, which hint to the existence of a possible body of about 10-times the mass of Earth. Calling this a discovery would clearly be premature. The confidence level is just a little above the "evidence" level of 3 sigma, under the assumption, that the discoveries of the KBO objects leading to the inference aren't observationally biased.
It's a long way to direct observation, since precise orbital data haven't been inferred.
Another option is, that there has been a planet. But it might have left our Solar system. Hence predicting a date for direct observation doesn't appear reasonable at the moment.