Solution:

Answer: $1010309$

Note that Niffy's favorite number must end in $9$, since adding $1$ makes it divisible by $10$. Also, the sum of the digits of Niffy's favorite number must be even (because it is equal to twice the number of digits) and congruent to $2$ modulo $3$ (because adding $1$ gives a multiple of $3$). Furthermore, the sum of digits can be at most $24$, because there are at most $12$ digits in Niffy's favorite number, and must be at least $9$, because the last digit is $9$. This gives the possible sums of digits $14$ and $20$. However, if the sum of the digits of the integer is $20$, there are $10$ digits, exactly $5$ of which are odd, giving an odd sum of digits, which is impossible. Thus, Niffy's favorite number is a $7$ digit number with sum of digits $14$.

The integers which we seek must be of the form $\overline{A B C D E F 9}$, where $A, C, E$ are odd, $B, D, F$ are even, and $A+B+C+D+E+F=5$. Now, note that $\{A, C, E\}=\{1,1,1\}$ or $\{1,1,3\}$, and these correspond to $\{B, D, F\}=\{0,0,2\}$ and $\{0,0,0\}$, respectively. It suffices to determine which of these six integers are congruent to $-1 \pmod{7}$, and we see that Niffy's favorite number must be $1010309$.