。
As of tomorrow it will have been raining for a week.
Literal
Tomorrow [with-で] 1-week rain [subject-が] continue-fall [progressive-ている] [nominalizer-こと] [dative-に] become.
降り続いている is the progressive form of the compound verb 降り続く ('continue raining'), shifting the aspect to 'has been continuously raining' rather than 'will continue to rain.' The result is a present-perfect-progressive flavor projected onto the 'as-of-tomorrow' frame.