The JPS translation is more faithful to the Hebrew wording of the MT here. The original word, כארי, means "like a lion". See H738 here.
The KJV may be following the understanding of some commentaries, such as Rashi (here in Hebrew), which understand this verse to mean that "my hands and feet have been attacked by a lion" i.e. they have been pierced by it's teeth.
(Note that in Strong's concordance linked above, the definition of pierced is noted, and sourced to the KJB margin, however, there were no marginal notes on this verse, so I don't know what motivated that.)
It is more likely that the KJV abandons the MT here, and follows a variety of other versions, as explained in the Cambridge commentary:
The A.V. here rightly deserts the Massoretic text in favour of the reading represented by the LXX, Vulg., and Syr., which have, they dug, or, pierced. Another group of ancient Versions (Aq. Symm. Jer.) gives they bound. (Fixerunt in some editions of Jerome is a corruption for the true reading vinxerunt.) The Massoretic text has, like a lion my hands and my feet. A verb did they mangle must be supplied, but the construction is harsh and the sense unsatisfactory. It seems certain that a somewhat rare verb form כארו (kâ’ărû), ‘they pierced,’ has been corrupted into the similar word כארי (kâ’ărî), ‘like a lion.’ The Targum perhaps preserves a trace of the transition in its conflate rendering, biting like a lion.
The alternative word would be found as H3738, which means dug, but could technically mean pierced.