I really liked this. It was more challanging than I orginally thought.
I do believe you could better explain the introduction of the letters into the mix. I think that if you gave an example using letters it would be very helpful. Something like the following:
When we get to 10!, the number could be 0 to 10, but since 10 is two digits, we start with A. So, 11! could range from 0 to B.
Example: 4790016000 would be coded as A000000000000 because 4790016000 (base 10) = 10x12! + 0x11! + 0x10! + 0x9! + 0x8! + 0x7! + 0x6! + 0x5! + 0x4! + 0x3! + 0x2! 0x1! + 0x0! and we 10 can't be used as a single digit.

Anyway, great kata!