In order to find $N$, which is the smallest possible integer whose digits have a fixed product, we must first find the minimum possible number of digits with this product. Once we have determined the digits that form $N$, then the integer $N$ itself is formed by writing the digits in increasing order. Note that the digits of $N$ cannot include 0, or else the product of its digits would be 0. Also, the digits of $N$ cannot include 1, otherwise we could remove the 1s and obtain an integer with fewer digits (thus, a smaller integer) with the same product of digits. Since the product of the digits of $N$ is 2700, we find the prime factorization of 2700 to help us determine what the digits are: $2700=27 \times 100=3^{3} \times 10^{2}=3^{3} \times 2^{2} \times 5^{2}$. In order for a non-zero digit to have a factor of 5, then the digit must equal 5. Since 2700 has two factors of 5, then the digits of $N$ includes two 5s. The remaining digits have a product of $3^{3} \times 2^{2}=108$. Therefore, we must try to find a combination of the smallest number of possible digits whose product is 108. We cannot have 1 digit with a product of 108. We also cannot have a 2 digits with a product of 108, as the product of 2 digits is at most $9 \times 9=81$. We can have a product of 3 digits with a product of 108 (for example, $2 \times 6 \times 9$ or $3 \times 6 \times 6$ ). Therefore, the number $N$ has 5 digits (two 5s and three other digits with a product of 108). In order for $N$ to be as small as possible, its leading digit (that is, its ten thousands digit) must be as small as possible. Recall that $N$ cannot include the digit 1. The next smallest possible leading digit is 2. In this case, 2 must be one of the three digits whose product is 108. Thus, the remaining two of these three digits have a product of $108 \div 2=54$, and so must be 6 and 9. Therefore, the digits of $N$ must be $2,6,9,5,5$. The smallest possible number formed by these digits is when the digits are placed in increasing order, and so $N=25569$. The sum of the digits of $N$ is $2+5+5+6+9=27$.