Alan, you are over stating the a) actual influence Rasputin had over Alexandra and b) the influence Alexandra had over Nicholas. As stated earlier, one must not necessarily assume just because it is written in a book that it is true or accurate.
When the history is actually reviewed for the facts and not the rhetoric,of the 26 ministerial appointments/dismissals that took place between 1914-1917 only 7 were actually in any way attributable to AF. The actual number out of these 7 times in which Rasputin made the suggestion or had some part in, to the Empress, which then went to Nicholas and which Nicholas actually agreed to was, a whopping, ONCE: Scherbatov, Minister of the Interior. One more, Samarin, head of the Orthodox Synod, Alexandra supported Nicholas' decision but did not actually instigate the idea or action. Two other men BLAMED Rasputin, but only after their replacement which became part of the "history" but in actuality there was no mention of them between Rasputin and the Empress ( Khostov, Minister of Internal Affairs, was unliked by Alexandra but not at any suggestion of Rasputin, and Polivanov- Minister of War who just blamed Rasputin with no actual substantive evidence to support the accusation).
THATS IT. Now the effect of Rasputin on the Empress may be a bit more than that, however, one must take the following into account: Two different investigations were undertaken on Rasputin years before the war. One at the request of Nicholas, the other undertaken by Stolypin and presented to Nicholas who had the report investigated for accuracy. BOTH TIMES, the reports were full of made up, salacious rumors, which overshadowed the "real" faults of Rasputin (ie: drunkenness and woman chasing. Both of which Nicholas tolerated because, to be blunt, most of the military men with whom he socialized did the same things)...
So, by the time of the War, both Nicholas and Alexandra simply wouldn't listen to anything more about Rasputin, because the Court and Ministers had all "cried wolf" about him too many times before. This part really can't be blamed on Alexandra, since all previous "reports" about Rasputin turned out to be false and made up lies to discredit him, she rather naturally believed the same later.