Hey Edward,
If you are looking at doing hiking in the Northern Redwoods there are a number of trails that can be recommended for you and your daughter that I know and have hiked that would work out for you and your daughter. I disagree with Ken that there are not any trails for what you are after.
If you are looking at doing hiking in the Southern Redwoods I would have to defer to the Bay area hikers.
Something to consider would be doing two different Northern Redwood routes. Each of them can be one or two nights on trail (each direction). This would allow you and your daughter to see slightly different zones of the Redwoods.
For example, you could do a yo-yo of the CCT at the Southern end of Crescent City, that could be one or two days and then bounce back, or get a ride back. You could then bounce down the coast about 10 miles and spend a couple of days in the Gold Bluff beach area, you could easily spend two or three days doing routes through this truly beautiful region.
Another option would be doing the CCT from Crescent City down to Prairie Creek Redwoods State Park, without breaking it into multiple car rides, this would give you a nice few days days on the trail, give you and your daughter a bit of beach hiking, a bit of inland Redwood hiking, and a stop or two in some neat little towns. An alternative would be to stop at Klamath, resupply and turn around and yo-yo it – or just catch a ride back to CC.
Anyway, just a couple of suggestions to throw out at you. Don't like to counter what Ken said above, but he is incorrect on this issue. Check out the options I mentioned above online and if any of them sound interesting I can give further insights on the options mentioned above.
** post edit update **
An additional option would be, if you want to get some miles on the Lost Coast Trail, yet still get in some Redwoods, would be to do the Northern half of the LCT (which as already pointed out, has no Redwoods) and then hike some connector routes over the Humboldt Redwoods State Park, which has some nice trails and spend two or three days in. The HRSP is nowhere near as nice as the Prairie Creek/Gold Bluff SP but it tends to be drier and a bit easier to access if you are driving from the bay area.
An additional option would be to just ignore NorCal all together and just do trails up in Oregon, of which greatly increases your options to hike in Redwoods. There are tons of routes up in OR to hike in the Redwoods. They have both coastal and inland Redwoods.