Clustering of documents using the topics derived from Latent Dirichlet Allocation
Assuming that LDA produced a list of topics and put a score against each topic for each document, you could represent the document and it's scores as a vector:

Document | Prime | Minister | Controversial | TopicN | ...
   1       0.041    0.042      0.041          ...
   2       0.052    0.011      0.042          ...


To get the scores for each document, you can run the document. as a bag of words, through a trained LDA model. From the gensim documentation:

&gt;&gt;&gt; lda = LdaModel(corpus, num_topics=100)  # train model
&gt;&gt;&gt; print(lda[doc_bow]) # get topic probability distribution for a document


Then, you could run the k-means on this matrix and it should group documents that are similar together. 
K-means by default is a hard clustering algorithm implying that it classifies each document into one cluster. You could use soft clustering mechanisms that will give you a probability score that a document fits within a cluster - this is called fuzzy k-means. 
 https://gist.github.com/mblondel/1451300 is a Python gist showing how you can do it with scikit learn.

ps: I cant post more than 2 links