How to calculate the concentrations of conjugate species present in the solution of diprotic acid?
You messed up calc in eq 2 in multiple ways...

$$K_1 = 0.017$$ 

$$K_2 = 10^{-7.19} = 6.5\times10^{-8}$$ 

$$K_1K_2 = 1.11\times10^{-9}$$

Also 

$$K_1 = \dfrac{\ce{[H^+][HSO3^-]}}{\ce{[H2SO3]}}\tag{1}$$

$$K_2 = \dfrac{\ce{[H^+][SO3^{2-}]}}{\ce{[HSO3^-]}}\tag{2}$$

$$K_1K_2 = \dfrac{\ce{[H^+][HSO3^-]}}{\ce{[H2SO3]}}\times\dfrac{\ce{[H^+][SO3^{2-}]}}{\ce{[HSO3^-]}} =\dfrac{\ce{[H^+]^2[SO3^{2-}]}}{\ce{[H2SO3]}}\tag{3}$$

rearranging (3)

$$ \ce{[SO3^{2-}]}= \dfrac{K_1K_2\ce{[H2SO3]}}{\ce{[H^+]^2}}= \dfrac{K_1K_2(0.010-0.0076)}{0.0076^2} = 4.6\times10^{-8}\tag{4}$$