Sorry that I misunderstood.

In addition to Tycoon's excellent instructions, the Monorail has express trains to Haneda from Hamamatsucho which can save some time. So take a look at the signs on the platform and decide which train you want to take.

As for the Keikyu line from Shingawa to Haneda, make sure you know whether you are boarding a direct train to Haneda or thru train to Yokohama and further south. In the later case, you need to change at Keikyu Kamata stn. to the Haneda train. Some early morning express from Shinagawa to Haneda can take as little as 15 minutes (before 6AM). But most directs take about 25 minutes.

Monorail has two stations at Haneda, one for JAL and one for ANA but Keikyu line use only one station.

As for taking a taxi, we took a taxi Friday a week ago from Haneda to Marunouchi and dropped off my luggage at Shinagawa on the way, it took over an hour in the rain and over Y6000.