Haneda to Kyoto

Hi,
Hyperdia will automatically try to give you the quickest route, but aside from not using the Nozomi the computer doesn't know if you are using a JR Pass or not.
Keikyu is not part of JR so you'd have to pay for that leg. Using the monorail is covered by the pass though. So for "best", you can decide if you fully want to squeeze more out of your pass, or pay for Keikyu and save a bit of time.