Does a propitiatory expiation (kaffarah) become obligatory due to the breaking of a wooden hand after it was made and installed, or has the oath already been fulfilled?

The determinant in specifying the intent of an oath is the intention of the swearer. If the intention is absent, reference is made to the cause of the oath. So, if your intention or the cause of your oath was to install a fixed hand at a specific time for its use, then what you did is not considered fulfilling the oath, and you must offer an expiation for breaking an oath.