Some problems decompose cleanly that way, some don't. It's not hard to design a system where
most is functional, though,
and that's still useful. You've probably done that without considering it "functional programming"TM, just trying to cut out global variables and whatnot. Decades ago, "structured programming" was the new thing, and manifestos appeared about how gotos were "considered harmful".
If you have a system where 95% of it is functional, cool, you've eliminated 95% of the places where you have to double-check for dataflow/mutability-related bugs. (To heavily paraphrase jerf's comment: functional programming is about accepting restrictions that make debugging simpler.) Make the system as clean as feasible, but don't get bent out of shape trying to get the last 5% FP-approved. It's probably less trouble to just know that's the hairy part.
The big idea in functional programming is that working with immutable data structures makes a lot of annoying bugs go away, and goddamnit, they're right. Does that mean you need to go 100% immutable? No - it's just a good default.
FP-style programming without (generational) garbage collection is really awkward, though - immutability means you're going to be creating a lot of temporary data structures. Good language implementations will recognize this and just stack-allocate them or let them go in the first collection.