Does an instrument exist to measure the different types of drag?
Yes, partially.  A wind tunnel's wake rake (an array of pitot tubes) can be adapted to measure in-flight profile drag (form drag plus skin friction).
Measuring Wing Profile Drag using an Integrating Wake Rake, by Pifer and Bramesfeld, in Technical Soaring 36(3):74-82, 2012.
The article doesn't have any in-flight photos, so it may not have actually been flown at time of publication.

@quietflyer mentioned Richard Johnson, who indeed has flown an airplane with a drag rake.  

The wing drag rake was installed on N3176T’s left wing trailing edge, 1.0 meters outboard from the wing root joint. ...

R. Johnson, &quot;A flight test evaluation of the SZD-55-1,&quot; Soaring, March 1992.
The drag rake itself is described and shown in another article:
R. Johnson, &quot;At last: an instrument that reads drag!,&quot; Soaring, October 1983.