What is the frictional force if you don't have the coefficient of friction but know that a downward force of 60 N at 10 degrees below the horizontal keeps the object moving at a constant speed?

If the speed is constant the frictional force must be equal to the propelling force (which is gravity in this case) as any imbalance of forces creates an acceleration.

If the 60N is in the same direction as the movement, then the frictional force is also 60N. If 60N is the gravitational force, you'll have to use the angle to calculate the propelling force that acts in the direction of movement. Again, the frictional force will be exactly the same as the propelling force in the direction of movement (only in magnitude, though - the direction of the frictional force (the vector) will be in the opposite direction).