Check which port each instance is running on in SQL Configuration Manager - SQL Server Network Configuration - TCP/IP - IP Addresses - IPAll - TCP Dynamic Ports

Connect using the IP address, port number (eg. 111.111.111.111,1433) and run:

select @@servername

sp_dropserver 'Old Server Name\Instance_name'

go

sp_addserver 'New Server Name\Instance Name','local'

go

Then restart SQL services for that instance including SQL Browser.