Say

Total number of sons = x and

Total number of daughters = y

"She has as many brothers as she has sisters"

number of her sisters = y-1

So

x = y-1 -- (1)

"Each one of her brothers has twice as many sisters as he has brothers"

number of his brothers = x-1

number of his sisters 2(x-1)

So

y = 2(x-1) -- (2)

Substituting y value from (2) in (1)

x = 2(x-1) - 1 = 2x - 3

That is x = 3 and y = 4

Hence

Number of sons = 3

Number of daughters = 4