Theory of Complexity – Prove the Existence of a Language $ L in DTIME (n ^ { log n}) $ that is not in $ Avg-P $

I have trouble with the following question:

To define $ P-Avg the class of all languages $ L $ for which there is a polynomial Turing machine $ M $ as for each $ n $, for all except $ frac {2 ^ n} {n} $ of $ 2 ^ n $ length ropes $ n $, $ M $ gives the correct answer as to whether $ x in L $ or not.
Prove that there is $ L in DTIME (n ^ { log n}) $ which is not in $ P-Avg.

My attempt: I thought about using diagonalization. First, we list all Turing machines $ M_1, M_2, M_3, … $. We now define the following MT: for the input $ x $ length $ n $simulate $ M_n $ entrance $ x $ for $ n ^ { log n-1} $ not. If the simulation stops, answer the opposite. Otherwise, reject. Now, let's define $ L (T) = L $. It's easy to see $ L in DTIME (n ^ { log n}) $. Suppose now that the contradiction $ L in P $ Avg. So there is $ M_i $ who gives the right answer to whether $ x in L $ for all except $ frac {2 ^ i} {i} $ of $ 2 ^ i $ length ropes $ i $. But because $ T $ simulated $ M_i $ and responds on the contrary, it is a contradiction.

The problem: I did not find a way to ensure that the simulation actually stops $ M_i $and responds on the contrary as wanted. Any ideas how can I solve this solution / solve in another way?