trade offs between number of features with its score
First of all, the Silhouette score comes always with visual inspection so be careful using it. The idea behind this score is having either Normally distributed clusters or having pretty compact clusters. If the intrinsic clusters within your data are not one of those, then this score is somehow meaningless.

Secondly, the question also needs a re-thinking. "Which features to include in the clustering?" ... well, all!

The characteristic of a dataset is more than just the combination of characteristics of each individual features. In these cases, usually Dimensionality Reduction algorithms are used, which reduce the dimensionality of data and extract/select informative features according to the entire feature set. Note that a feature that you might exclude, makes a meaningful partitioning in combination with some other features.

at the end I recommend having a look at K-Modes which is designed for clustering datasets with nominal attributes. Most probably helps you more than k-mans.

If you like to get some insight about Graph Clustering (which is also related to your problem and is one of typical solutions) you can comment here. 

Good Luck!