You are applying the raising operator to the maximal state of angular momentum. Perhaps that is the reason you are not getting the eigenvalue.

In fact, applying the lRaising to the minimal state with the eigenvalue -l:

In[107]:=Solve[Simplify[Lz[(x - I y)^l] == eigenvalue (x - I y)^l],eigenvalue][[1]]
         Solve[Simplify[L2[(x - I y)^l] == eigenvalue (x - I y)^l],eigenvalue][[1]]

Out[107]= {eigenvalue -> -l \[HBar]}
Out[108]= {eigenvalue -> l (1 + l) \[HBar]^2}

does increase the eigenvalue to -l+1:

In[109]:=Solve[Simplify[Lz[lRaising[(x - I y)^l]] == eigenvalue lRaising[(x - I y)^l]],eigenvalue][[1]]
         Solve[Simplify[L2[lRaising[(x - I y)^l]] == eigenvalue lRaising[(x - I y)^l]],eigenvalue][[1]]

Out[109]= {eigenvalue -> -((-1 + l) \[HBar])}
Out[110]= {eigenvalue -> l (1 + l) \[HBar]^2}