Hello K2,

I am using the Content Module to list K2 Items in the sidebar. My goal is to filter the items by Category (in alphabetical order) and then list each Category's items underneath each Category name (in alphabetical order). Basically, each category name is listed once and its items are listed underneath like a typical e-commerce product sidebar.

What do I need to do to accomplish this? Thanks in advance!

I would use several K2 Content modules, one for each category.
These modules would pull x items from one category only and in the module's settings I would select alphabetical ordering.

Actually, that still leaves the formatting the same because the Category is listed under each Item name. Which is exactly what I don't want. Is there a way to accomplish my initial request with only ONE module?

With one default module No.
You can hide the category from the module's settings.

Forgive me for not being clear, I don't think you understand what I am trying to achieve. I don't want to hide the Category. I want to list each Category and then list the Items underneath like so:

Category A
Item 1A
Item 2A
Item 3A

Category B
Item 1B
Item 2B
Item 3B

Is it possible to manipulate the settings or edit the PHP to obtain the result above?

You do not need to manipulate anything.

Simply publish separate K2 Content modules. One for each category.
Then sort these items alphabetically (module settings).

The title of the module can be the either the category or you can hide it and use the custom link as the module (link it to the menu item).
If you choose the latter then you might need to tweak the template a bit so the custom link appears at the top.

That is what I have done as a temporary work-around but unfortunately my client will not be satisfied with the layout. Is there a way to accomplish the same layout but within one module?

As I have already told you it is not possible.