This isn't as easy as you might think.
Each process has a "Responding" property, but all processes that have a "MainWindowHandle" (that isn't a zero) have a window and all processes that have a window always report that they're "Responding". So unless your application is a console application that's useless.
If you're looking for "zombie" or ("orphaned") processes (ones that have a ParentProcessID) but the parent process has ended, then that's usually pretty easy: for each process, the ParentProcessID should exist as a process. You can't use Get-Process for this, but Get-WMIObject or Get-CIMInstance will report the ParentProcessID (using the Win32_Process class).
But if you're looking for "idle" processes you'd have to keep track of each process (or maybe just ones with "ProcessName" that's the same as you application -- minus the file extension). You'd also have to separate the "user" time and "system" time. If both are non-varying between checks then they might be candidates to be killed. You may also be able to check the owner of each process and if the owner is no longer present on the server the process may be a candidate for termination. You'd have to update the information and add new processes and remove processes that have ended normally.