Best Metric to Compare Sparsity of Vectors
I am sorry I cannot comment your answer due to my low reputation. Gini and your suggested sparsity ratio ($l_1(x)/l_2(x)$) both give me the same value for $\lambda$. But 


The problem I still see is that I cannot take into account how well the vector is solving the equation $Ax-y$. I would like to combine the residuum $l_1(A\hat{x}-y)$ and the sparsity $l_1(\hat{x})/l_2(\hat{x})$ into a new metric value. Do you know something from the literature?
Gini only works with positive values, but my measurement vector has lots of negative values. In my eyes, this is a heavy disadvantage because many sparse vectors can have negative values.