Lowes, sells a tool, called a mud stomper, that will do that texture for you. It is a rounded brush type thing and comes with a short handled pole, but you can buy a longer handle and do it all from the floor, if necessary. It works very easily. Your texture, looks a bit less than what mine is, so your mud is put on a bit lighter than mine. The tool, is what makes the pattern on your sheetrock. You cant tell, the repair from what was there, on mine. You can find directions for using it online. Good Luck

It could be that the wire (expanded metal) was only a corner reinforcement. I usually pull the rest of it out if that was the case, it is just a source of trouble going forward. It's possible that they wire-lathed the whole ceiling, but it's not likely. For patches like this I spray bonding agent (Link or Plasterweld & other brands) Wipe off any drips of Link, they are hard to get rid of when dry. Then use a setting-type drywall compound (aka Hot Mud; Durabond 90-minute or 45-minute set time) Which dries faster and with less shrinkage than mud that simply sets by evaporation. Don't try to fill deep/wide gaps in one shot. Once you have the surface filled level, use a wide fiberglass repair tape over the repair zone, and then you can skimcoat with normal green-lid mud after bedding the mesh in the Durabond. Hot mud cures very hard, it has some Plaster of Paris in it, so it is very hard to sand, but ridges can be scraped off. The bonding agent really does the trick here. I have some drywall/plaster ceiling joints that have not re-cracked over ten years. FWIW, Link is close to the same stuff as white elmers glue, a PVA adhesive. Casey

I live in Portland,Oregon where texture is the norm on newer housing stock. A friend of mine, who is also from the Mid-West and a professional painter, hates texture. He has methodically gone through his whole house skim coating the walls with drywall compound compound. His wife thinks he is nuts! I personally don't mind the appearance of textured walls, but they are a pain in the rear to patch and touch up well, much harder than flat wall. Also, if you are a fan of wall coverings or faux finishes, they are also a hindrance.

I can't see the texture in that photo -- need something much closer. In the meanwhile, here's an image search on "drywall textures". Find the one(s) that are closest to what you have and then search for possible methods to create it.