Lets put some physics to play (mostly because I want to do some numbers....) Assume the brakes did all the work.....
200kg all up weight of rider, bike, trailer and kids. 350Meter vertical descent means you need to dissipate 700kJ of energy = 200WH.
I found a reference to "Road and mountain brakes must absorb 75 watt-hours of energy over a 15-minute period without failure to be legal for sale in Europe" so will use it for my calculations. 200WH at 75WH/15Min is 40 Minutes.
So assuming you spend 40 minutes doing the descent not to exceed the 75WH/15m limit of the brakes, and the the angle of the slope is 15 degrees, 350Mters is about 1.5km - you could safely descend the hill at no more than 2km/hour -seems too slow. Maybe that standard is 75WH/15m per brake - so you could twice as fast 4km/h. Even if other friction, tire scrubbing and aerodynamic losses (At that speed ?) were 1/2 of the energy loss - you cannot exceed 8km/h without over heating the brakes. But a cold set of brakes stores energy as heat - so what is the acceptable rate of energy dissipation after 15 minutes?
As others have said, you boiled the fluid - replacing it is not going to fix the problem. If you do this again with the kids.... "Your gonna need a bigger brake"... duh da, duh da.... (...Movie reference...)
In researching this answer I am now far more aware of limitations of my brakes - thanks....