Hello everyone

In connection with calculating the Fourier coefficients of some quasi-modular forms which I have been looking at lately, I have come across the following type of sum

$$ S_{a,b}(N) := \sum_{t=1}^{N-1} \ \ \sum_{(n,m) \in I_{N-t,t}} \frac{1}{m^a n^b} $$

where $$ I_{k,l} = \{ \ (m,n) \ \big| \ \ m|k \ , \ n| l \ , \ m>n \}$$
Note the final condition in $I_{k,l}$ which is of course what prevents this sum from simply being a sum of products of divisor functions. My questions are

1.) Can anybody think of a way of calculating this sum rapidly for large $N$. Perhaps by somehow expressing it as a sum of some modified divisor functions or something similar.

2.) Has series of this type occurred elsewhere in math?