I have not used the taxi services in Moscow in a couple of years, but the following site would be a good reference.
http://www.go-to.ru/en/catalog/transfers/index.php
I had a lovely stay at the National a couple of years ago. Did not have the Kremlin view, but lovely room and great breakfast. Interesting, I just stayed at the Park Lane and was so cold in October. There was no heat and it was cold outside. They sent a space heater to my room, but it was very slow to warm up the room. I was not very comfortable.
I must say I love the hot Moscow rooms, which are not uncommon in Russia. Personally I prefer that to freezing, but frankly, one should be able to adjust the room temperature to one’s liking.