Word for - when a tool becomes an extension of the body
I first thought élan might be the word you're looking for, as quoted from John Muir's "How To Keep Your Volkswagen Alive":
... with the Mechanical type, you must keep your foot on the accelerator with verve, élan and confidence, as it won't forgive inattention
but the definition didn't match the request. From Merriam-Webster and others:
élan: Vigorous spirit or enthusiasm; Enthusiastic vigor and liveliness; Distinctive style or flair.
Then I thought finesse might fit the bill, and came across prowess in the Thesaurus. Again, from Merriam-Webster:
finesse: Refinement or delicacy of workmanship, structure, or texture; Skillful handling of a situation : adroit maneuvering
prowess: distinguished bravery; especially: military valor and skill; extraordinary ability "his prowess on the football field"
Ah, but there in the definition of 'finesse' - the word adroit, with the usage being adroitness. From Dictionary.com:
adroit: Expert or nimble in the use of the hands or body; cleverly skillful, resourceful, or ingenious
Hopefully that is the word you're looking for.
[http://www.merriam-webster.com/dictionary
http://dictionary.reference.com/browse
Muir, John, and Tosh Gregg. How to Keep Your Volkswagen Alive: A Manual of Step by Step Procedures for the Compleat Idiot: 1200, 1300, 1500, 1600, 1700, 1800 & 2000. 16th ed. Santa Fe, NM: J. Muir Publications, 1979. Quote from Chapter 10; subsection "Rap On Timing"]