I have similar situation in my office to trace down the particular machine.
Here is what I did on 35xx XL switch(IOS base). Just to get idea.
At the router which is gateway of that machine, look up the mac address of machine( In cisco router "sh arp" or newer ios version support "|" output modifier so you can use "sh arp | include x.x.x.x"
In the switch(first link to router)," sh mac-address-table | inc <mac address>"
if it show Fastethernet port, make sure that port is not trank port by doublechecking "sh mac-address-table | inc <fastethernet x/x>. You should only see one address from access port. If it is Gbit port or FA trunk port, follow the next switch. Of course, you have to know which switch link to which.
I think 19xx switch has different command set.
I never use 1900 series so I can't really say which command.