If i got it right, i first put a number, can be any number, in front of HNO3. Because I2 can have any number in front of it because it only affects the value of the number iodine atoms on the reactants so i solved that number last. Then I put that same number in front of NO2 because that way the number of nitrogen atoms are balenced. Then I realized how many oxygen atoms need to be balanced on the products. Which is the number in front of HNO3, then given how many oxygen atoms need to be balance i guess-and-checked until the oxygen and hydrogen atoms balanced. Then i fould the number in front of I2. Finally, i found the least common multiple so the numbers are whole numbers. For example:
.5 I2 + 5 HNO3 -> 1 HIO3 + 5 NO2 + 2 H2O, would become I2 + 10 HNO3 -> 2 HI)3 + 10 NO2 + 4 H2O