Your understanding of the notation is incorrect. The quantity dA^{[L]} is a vector. There is no average taking place there, which is the only point at which the factor of \frac {1}{m} would be necessary. The only sums there are within each element of the vector. There are commas not + signs between the entries, although perhaps that’s a bit hard to see in the slide.

This is a common confusion with the way Prof Ng uses his notation. It is a bit ambiguous and you have to understand the context to know what quantity is in the numerator of the partial derivative. Here’s another very recent thread on this general point.