The only way to specify the mimetype in Alfresco Public ReST API is by uploading content and making sure that the file parameter in the reqeuest body already has the proper mimetype specified. Changing just/only the mimetype afterwards / at any time is not a change for which Alfresco provides a direct API - not even when developing in-process extensions in Java.
Looking at the source code of the implementation for the ReST API I can clearly see one hugh reason which might cause the result to not show what one might expect. The whole operation is contingent on the node path, which might be changed by moves / renames, and on any such changes, any previous audit entries will no longer be accessible. This might not affect your particular case right now, but in general, this operation has been recognized as "broken", and there is even an issue to deprecate it. One of our Order of the Bee members, @yreg actually got the ball rolling on this, which is why I remembered the issue as soon as I looked at the code. It obviously is a bit of a disappointment that they have not yet done the simplest thing with regards to deprecating that API or at least listing the issue in the api-explorer documentation in all the time the issue has been open.