I feel like there's a lot of confusion here. Perhaps it is my confusion, but I need to make sure everything is crystal clear here so I'll lay it all out.
I've downloaded the patch zip file and extracted the patch. Where do I put it on my player?
You don't. Putting the patch .zip on the device will not achieve anything.
You need to
set up a development environment,
build the crosscompile toolchain for your target architecture,
checkout the Rockbox source code repository,
apply the patch to the source, and then finally
compile a Rockbox binary for your target.
Then you can copy the patched build that you have compiled to your device, at which point you will lose support (unless you can replicate the issue on an official build) due to having a modified build (as you have now) and in order to receive further updates in future you would need to continue to compile your own binaries as updating to the official builds will remove the patched behavior (as you discovered).
What isn't clear here is if you are talking about a Rockbox binary archive, or a patch archive. If you are indeed talking about an archive that contains the line out patch, then you would need to refer to the beginning of this post.
[Saint]