Keep gutters clean, properly sloped and running freely. Many Mansards don't have gutters between the two roof planes, but if yours does, they may be the cause of your icicles.
Insulate your attic under the roof so that it meets local R-value standards. This prevents heat from the house from quickly melting snow, which supplies the water that creates the ice dam and the leak.
Ventilate your attic to keep the roof cold so that ice is less likely to melt and form dams. If natural, even ventilation from the eaves is not possible, a fan facilitates circulation. A thermostat causes the fan to turn on when the roof gets too warm.
Install snow guards on the roof to keep snow from sliding down your Mansard roof and collecting at the base of the roof plane.