IDA Free doesn't support the "Apply patches to input file..." feature. Hence, you'll have to do this the old way. I'll expand my answer and will go over things you already know so others (e.g coming from search engines) can benefit from a whole complete answer.
Edit the configuration file
The first thing you have to do is to modify an IDA GUI configuration file named idagui.cfg. You should be able to find the file at "IDA Free\cfg\idagui.cfg". Locate the file and change "DISPLAY_PATCH_SUBMENU" form "NO" to "YES".
After that, start IDA Free and you'll see a fresh new sub-menu called "Patch program".
Patching
You should be able to use this sub-menu to edit the program. It's pretty intuitive, just put your cursor wherever you want to make a change, and choose the appropriate option from the sub-menu.
Notice that every patch you do is only affected on the IDB, the IDA Database. Thus, the binary on disk won't be affected and when you'll try to debug/execute it you will not see your patches.
Exporting a .DIF file
When you finish with all your patches, it's time to apply them to the binary on disk. To do this, you'll first have to produce a .DIF file that will contain a list of the changes you've made.
To produce this file go to File -> Produce File -> Create DIF file...".
Applying the changes to the binary
You can use this Python script by stalker to apply the changes. Use it like this:
$ idadif.py <original_binary> <IDA_DIF_file.dif> [revert]
Alternatives:
Note that I didn't test these alternatives and I'm pretty sure that some of them will not work with IDA Free
