I would go to the Sheraton Miyako.
You should be still able to get on the monorail to Hamamatsucho then Taxi to hotel OR Yamanote to Shinagawa then taxi to hotel.
You can have breakfast free as a Plat then get on the early shuttle to Shinagawa station...then Yamanote back to Hamamatsucho and monorail back to HND.
Have fun!