How to get the probability of belonging to clusters for k-means? I need to get the probability for each point in my data set. The idea is to compute distance matrix (first column contsins distances to first cluster, second column conteins distances to second cluster and etc). The closest point has probability = 1, the most distant has probability = 0.
The problem is linear function (like MinMaxScaller) have output where almost all points have almost the same probability.

How to choose nonlinearity for this task? How to automatizate this process on python? For example for the most closest point p=1, for the most distant point that belongs to cluster p=0.5, for the most distant point p is almols 0.

Or you can propose another methods for computing this probability.
Let us briefly talk about a probabilistic generalisation of k-means: the Gaussian Mixture Model (GMM).

In k-means, you carry out the following procedure:
- specify k centroids, initialising their coordinates randomly
- calculate the distance of each data point to each centroid
- assign each data point to its nearest centroid
- update the coordinates of the centroid to the mean of all points   assigned to it
- iterate until convergence.

In a GMM, you carry out the following procedure:
- specify k multivariate Gaussians (termed components), initialising their mean and variance randomly
- calculate the probability of each data point being produced by each component (sometimes termed the responsibility each component takes for the data point)
- assign each data point to the component it belongs to with the highest probability
- update the mean and variance of the component to the mean and variance of all data points assigned to it
- iterate until convergence  

You may notice the similarity between these two procedures. In fact, k-means is a GMM with fixed-variance components. Under a GMM, the probabilities (I think) you're looking for are the responsibilities each component takes for each data point.

There is a scikit-learn implementation of GMM available if you wanted to look into that, but I'm guessing you just want a quick way to amend your existing code, in which case, if you're happy to assume your clusters are fixed-variance Gaussians, you could transform your distance matrix element-wise as $y = e^{-x}$ (giving you an exponential fall-off), and then calculating the softmax over your columns (normalising your distribution so $P(Y=1) + P(Y=2) + ... + P(Y=k) = 1$).

It's worth pointing out that the assumption your clusters are fixed-variance Gaussians isn't necessarily valid. If your dimensions have wildly different scales, this may produce strange results, as dimensions with smaller-magnitude units will appear more "probable". Standardising your data before running your clustering procedure should remedy this.