Rate of Mass Loss from the Solar Wind
$$\left( \frac{4 \pi (6.96e10\,\text{cm})^{2}}{1} \right)$$


This is the primary source of your error. Your value of 6.96×1010 cm is the radius of the Sun. The problem specifically said "Assuming at the Earth ...". You need to calculate the flux through the surface of a sphere whose radius is about one astronomical unit rather than one solar radius. The astronomical unit is 149597870700 meters (exactly), or about 1.5×1013 cm. This error alone makes your value low by a factor of about 50000. The remaining factor of two results mostly from using 10-24 grams per amu.

Dimensional analysis can only take you so far. While your result is dimensionally correct, you didn't think enough about the nature of the problem.