Nate,
Your morning walk is IMO a long one. How old are the children? Allow a lot of time for this morning walk, or maybe take the hop-on hop-off boat "Batobus" on your way to the Eiffel Tower to see things along the way. Your nighttime dinner cruise will have a different perspective than a daytime boat ride. Check out a Paris map for distances, etc. to allow enough time for this leg. Plan time to stop and check out things that catch your eye. (By the way, I am assuming you are just planning to see the EiffelTower from the ground, and not going up. To ascend, allow lots more time.)
Take the metro from the Eiffel Tower area to Le Marais. Schedule enough time for lunch and a sit-down sometime in there, after a morning of walking. For that night, take the metro or taxi to Montmartre. Note that in Montmartre most metro stops are long flights of stairs below ground, so if the children are small, allow time to get up top to street level.
Just fyi, if you decide on a taxi, get the G7 app on your phone to "call" one. Be sure to tell them 4 passengers. We have had 4 adults in one taxi with the driver having one of us in the front seat. But not all will do so. Taxis can travel in the bus lanes so are faster than Uber and from what I hear, don't cost much more (Uber surge pricing). Of course, the metro is cheaper than taxis, by far. It is up to your tolerance for steps and for crowds on the metro. Also, the age of your children (Five, ten, fifteen years old?) will be a factor.
Your one day sounds VERY busy with the long walk, but you all may be happy to bustle along. Have a fun day in the City of Light!