Patching paper towel dispenser in bathroom?

Ok so what's the question?
Remove it, patch the wall.

Question is how do I patch a hole that big

Got a picture?
Most likly there's going to be studs visable once it's out, just cut the sheet rock back enough so expose 1/2 the stud, cut patch, screw in place with sheetrock screws, apply drywall compound with a 6" wide drywall knife then go over the seams with drywall tape and apply another thin layer of compound. Let it dry, wipe over it with the knife and apply another thin layer of compound.
It will take at least three layers to get it to the point you can sand it.
Now prime and paint two coats.

As stated before, remove, patch the hole. If you do not wish to make it bigger, you can attach a piece of 1/2" plywood on either side for you to secure the patch to.

You could always try a hot patch out of DW, looks like a good candidate for one but if you've never done one before it probably would backfire on you.
take a piece of drywall about 14" x 14" measure hole area and cut away about 2" of dw all way around on backside leaving the paper in tact, what is left will be the area that fits into space, mud area on outside of hole and place the patch in the hole wiping excess mud off making sure all is flat and smooth as possible, let dry and keep applying mud until finished. like I said, someone who has done this as many times as I have its easy but a first timer ?? its not that hard really and its a quick fix.

personally, with that much space i would convert it into something else, like a shelf or something.

To add a drywall patch, cut the drywall back to the studs flush and add a cripple stud to each stud and possibly a header top and bottom. This will give you a good surface to work with for the drywall patch without fasteners being close to the edge.