To solve this problem, we need to follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).

Let's first convert all mixed numbers into improper fractions:

5 1/6 = 31/6
3 1/4 = 13/4
2 1/6 = 13/6
2 3/5 = 13/5

Now, we can substitute these values back into the equation:

31/6 ÷ 31 + (13/4 + 13/6) ÷ 13/5 - 2/3 ÷ 4/9

Next, we simplify the expressions within the parentheses first:

(13/4 + 13/6) = (39/12 + 26/12) = 65/12

Now, substitute this back into the equation:

31/6 ÷ 31 + 65/12 ÷ 13/5 - 2/3 ÷ 4/9

Next, we simplify the division:

31/6 ÷ 31 = 31/6 ÷ 31/1 = 31/6 1/31 = 1/6
65/12 ÷ 13/5 = 65/12 5/13 = 325/156
2/3 ÷ 4/9 = 2/3 * 9/4 = 6/12 = 1/2

Now substitute these values back into the equation:

1/6 + 325/156 - 1/2

Next, find a common denominator to combine the fractions:

1/6 + 325/156 - 78/156

Now, subtract the fractions:

1/6 + 247/156

Now, add the fractions:

247/156 + 26/156 = 273/156

Finally, simplify the fraction:

273/156 = 91/52

Therefore, the solution to the given expression is 91/52.