Is it a double-sided sprocket, or is the 15t the single-sided one?
Meh...... well, I have swapped my sprocket around at 12,000kms I think (17t) and will replace everything at ~24,000kms. So that will be the sprocket worn out on two sides. Your sprocket *should* last more than 5,000kms, but it really depends on what the chain is like. Place the chains together and measure 12" (24 links). If the old chain has extended to 12 1/8" or more for 12" of original links, the sprocket will be worn to the point that a new chain will skip. Your old chain doesn't skip because it is worn to fit the sprocket.
Oh, yes, the original chain - KMC Z50. A quick search and glance at the pictures, this looks to be a fairly budget-oriented chain. I would not be surprisded that it wears out (and the sprockets) at a quicker rate.
My prediction, as I said, is that a new sprocket will run fine