In case anyone is interested in taking this route, the 2060 airport bus (CitiAir company) runs between TPE and City Hall Bus Station, which is next door to the W Taipei. Fare is 145 TWD, which is considerably less than taxi fare (which isn't all that expensive to begin with).
Ride will be ~60 min +/- traffic. Comfort level: slightly above backpacker/hostel. You get what you pay for.