The function N usually evaluates with the standard
$MachinePrecision
If you insist in higher precision e.g. like this:
N[((-5551459608 + 3925475120 Sqrt[2]) Sqrt[2 - Sqrt[2]])^2, 100]
and
N[Expand[((-5551459608 + 3925475120 Sqrt[2]) Sqrt[2 - Sqrt[2]])^2], 100]
you get the same result. You could also try
N[((-5551459608 + 3925475120 Sqrt[2]) Sqrt[2 - Sqrt[2]])^2 - (210443293513022896768 - 148805879898289354304 Sqrt[2]), 100]
and study the warning. This link might help.
Cheers,
Marco
PS: Note that
Simplify[((-5551459608 + 3925475120 Sqrt[2]) Sqrt[2 - Sqrt[2]])^2 - (210443293513022896768 -
148805879898289354304 Sqrt[2])]
uses "infinite precision" and vanishes - as expected.