Display category posts, sorted by category, need to sort the category first

Hello
After searching around I found this code to display a list of all posts in, say Category 3, along with all the child categories of Cat 3, sorted by category. It’s working just the way I need it to.

However I need to have the category output sorted by the menu order as shown in the “Categories” tab.

I image that I need to take the “$cat_children” variable and sort it. For the life of my I can’t find how to do that. An “orderby=menuorder” of some sort?

Here’s the code:

[Code moderated as per the Forum Rules. Please use the pastebin]

thanks!