It's actually a Hyundai motor. On the bottom of my 2015 there is a little placard/badge that says Hyundai. So that may expand your search. You can source a motor from another turbo forte, veloster turbo, or maybe kia soul turbo. maybe others too.
I THINK that there was an internals upgrade sometime in 2013 or 2014. So I would try and find something that is 2014+ if you can.
You could also do a rebuild but that may be more work. - Kdm Tuners 1.6 Gamma G4fj Engine Build Kit - Pierce Motorsports
It does seem like its a little tricky to find an engine though... hope that helps.